#7501097

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500695
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35660	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2703 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=dfi,e^c=ei,g^c=gi,g^d=gi

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, 7408293);