#6811262

order	512
nilpotency class	3
isoclinism	isoclinic class of #6807031
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	#25440	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1724 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=hi,d^2=f,e^2=fhi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,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, 6718458);