#6816108

order	512
nilpotency class	3
isoclinism	isoclinic class of #6814815
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

2041 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)(3,6)(4,5)(7,8)(9,10,14,19)(11,12,23,17)(13,20,29,35)(15,32,16,21)(18,26,33,25)(22,24,39,28)(27,40,34,36)(30,38,31,37);(2,5)(3,7)(9,11)(10,17)(12,19)(13,27)(14,23)(15,18)(16,33)(20,36)(21,26)(22,30)(24,37)(25,32)(28,38)(29,34)(31,39)(35,40);(9,12)(10,18,19,33)(11,21,23,32)(13,28)(14,17)(15,25)(16,26)(20,27,35,34)(22,38,39,37)(24,29)(30,36)(31,40);(2,5)(6,8)(10,19)(12,17)(13,29)(21,32)(22,39)(25,26)(27,34)(30,31);(1,3,4,7)(2,6,5,8)(9,13,15,30,14,29,16,31)(10,20,32,38,19,35,21,37)(11,22,33,27,23,39,18,34)(12,24,25,40,17,28,26,36)

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,d^c=di,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, 6723304);