#72293192

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#1781
commutator	C₂×C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄×Q₈	C₂⁴
prime radical	C₂×#2397	C₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#556 ◁ G
LECS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

4457 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^10|c^2=e,d^2=hi,e^2=j,f^2=hj,g^2=j,c^a=cf,c^b=ch,d^a=dhi,d^c=dg,e^a=eh,e^d=ei,f^a=fhj,f^c=fj,f^d=fj,g^c=gij,g^e=gj,i^c=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72293197
maximal normal subgroup of	#72293197

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, 3, F.5);;
gap> SetPower(coll, 4, F.8*F.9);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 6, F.8*F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);