#7145765

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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

2177 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)(3,4)(7,11)(10,12)(13,14,17,23,18,22,30,36)(15,24,27,35,26,20,33,19)(16,21,31,40,32,39,41,44)(25,38,28,42,29,37,43,34);(6,9)(7,11)(13,15,18,26)(14,19,22,35)(16,28,32,43)(17,33,30,27)(20,36,24,23)(21,38,39,37)(25,41,29,31)(34,44,42,40);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,29)(17,27)(18,26)(19,23)(21,38)(22,24)(25,32)(28,41)(30,33)(31,43)(34,44)(35,36)(37,39)(40,42);(6,9)(10,12)(13,16,30,41,18,32,17,31)(14,21,36,44,22,39,23,40)(15,25,33,43,26,29,27,28)(19,34,20,37,35,42,24,38);(5,7,8,11)(6,10,9,12)(14,22)(15,26)(21,39)(23,36)(25,29)(27,33)(28,43)(40,44)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=fi,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7052961);