#72285656

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#1879
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	#7623099	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ #20473 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1874 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,4)(5,12)(6,9)(7,8)(10,13)(11,14)(15,16)(17,18,23,28,37,44,34,45,21,27,22,38,24,29,35,48)(19,30,32,43,40,46,36,25,31,42,20,26,33,47,39,41);(1,3)(2,7,9,15)(4,12,14,10)(5,8)(6,11)(13,16)(17,19,21,31)(18,25,27,41)(20,34,32,35)(22,39,23,36)(24,40,37,33)(26,44,43,29)(28,47,38,46)(30,45,42,48);(1,4,13,15,6,14,5,7)(2,8,10,3,9,16,12,11)(17,20,35,19,24,36,22,33,21,32,34,31,37,39,23,40)(18,26,45,42,29,25,28,46,27,43,48,30,44,41,38,47);(2,9)(3,11)(8,16)(10,12)(18,27)(25,41)(26,43)(28,38)(29,44)(30,42)(45,48)(46,47)

Polycyclic presentation:

2^10|a^2=g,b^2=e,c^2=efhi,f^2=hij,g^2=ij,i^2=j,b^a=bf,c^a=cfghij,c^b=cf,d^a=dhj,d^b=dh,e^a=eh,e^c=eh,f^a=fhj,f^b=fi,f^c=fh,g^b=gij,g^c=gj,i^b=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285671
maximal normal subgroup of	#72285671

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.7);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.5*F.6*F.8*F.9);;
gap> SetPower(coll, 6, F.8*F.9*F.10);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);