#7503925

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500908
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1862 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=f,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,e^b=ei,h^b=hi,g^c=gi,h^c=hi,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, 7411121);