#7007251

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996799
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1952 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)(3,5)(4,6)(7,8)(9,10)(11,21,24,36)(12,27,23,39)(13,31)(14,20)(15,19)(16,25,37,26)(17,30,35,29)(18,32)(22,38)(28,34)(33,40);(1,2)(3,7)(4,6)(5,8)(9,11)(10,16)(12,28)(13,30)(14,26)(15,24)(17,38)(18,27)(19,37)(20,36)(21,40)(22,23)(25,33)(29,32)(31,39)(34,35);(1,3)(2,5)(4,8)(6,7)(9,12,15,23)(10,17,19,35)(11,22,24,28)(13,25,32,26)(14,29,33,30)(16,34,37,38)(18,21,31,36)(20,39,40,27);(3,8)(5,7)(9,13,14,22,15,32,33,28)(10,18,20,34,19,31,40,38)(11,23,25,30,24,12,26,29)(16,35,21,27,37,17,36,39);(2,6)(5,7)(10,19)(11,24)(13,32)(17,35)(20,40)(22,28)(25,26)(27,39)

Polycyclic presentation:

2^9|a^2=fgi,c^2=i,d^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6914447);