#7502475

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500689
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1788 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,d^2=f,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, 7409671);