#7503672

order	512
nilpotency class	3
isoclinism	isoclinic class of #7501009
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	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1898 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,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, 7410868);