#72319506

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator	C₂×C₄	C₂³×C₄²
Frattini, cosocle	C₄×C₈	C₂⁵
socle	C₂²	C₂³×#97
prime radical	#3444	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2480 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^10|b^2=g,c^2=f,d^2=j,f^2=h,g^2=i,i^2=j,b^a=bj,c^a=chj,c^b=chi,d^a=dhj,d^b=dj,d^c=dhj,e^b=ehj,e^d=eh,f^b=fj,g^c=gj

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319579, #72319792, #72343205
maximal normal subgroup of	#72319579, #72319792, #72343205

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