#6816106

order	512
nilpotency class	3
isoclinism	isoclinic class of #6814805
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

2051 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)(3,6)(4,5)(7,8)(9,10,13,18)(11,21,24,16)(12,19,27,34)(14,17,15,20)(22,32,25,31)(23,37,39,30)(26,40,38,35)(28,33,29,36);(2,5)(3,7)(9,11)(10,16)(12,26)(13,24)(14,25)(15,22)(17,32)(18,21)(19,35)(20,31)(23,28)(27,38)(29,39)(30,33)(34,40)(36,37);(10,17,18,20)(11,22,24,25)(12,27)(19,36,34,33)(23,38,39,26)(28,29)(30,37)(35,40);(2,5)(6,8)(10,18)(12,27)(16,21)(17,20)(23,39)(26,38)(28,29)(31,32);(1,3,4,7)(2,6,5,8)(9,12,14,28,13,27,15,29)(10,19,17,33,18,34,20,36)(11,23,22,26,24,39,25,38)(16,30,31,35,21,37,32,40)

Polycyclic presentation:

2^9|a^2=i,c^2=gh,e^2=fh,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,e^c=ei,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, 6723302);