#7501577

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500736
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	C₂×#3062	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1956 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,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, 7408773);