#7501089

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500689
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

2717 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)(21,30)(25,34)(26,35)(28,38)(31,36)(33,37)(39,40)(41,44)(42,43);(1,2)(3,4)(5,7,8,12)(6,9,10,11);(2,4)(13,15,19,28)(14,20,23,35)(16,25,31,41)(17,27,18,26)(21,34,37,44)(22,38,32,24)(29,39,36,43)(30,42,33,40);(6,10)(7,12)(13,16)(14,21)(15,25)(17,30)(18,33)(19,31)(20,34)(22,36)(23,37)(24,39)(26,40)(27,42)(28,41)(29,32)(35,44)(38,43);(13,17,19,18)(14,22,23,32)(15,26,28,27)(16,30,31,33)(20,24,35,38)(21,36,37,29)(25,40,41,42)(34,39,44,43)

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=df,e^c=ei,g^c=gi,g^d=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7408285);