#7064545

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1851 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=fgi,c^2=h,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=ehi,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6971741);