#7062508

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
prime radical	#42995	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1874 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfi,e^b=eh,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, 6969704);