#6816452

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

1674 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)(3,7)(4,6)(5,8)(9,10)(11,20)(12,17)(13,29)(14,18)(15,19)(16,23)(21,33)(22,34)(24,30)(25,28)(26,32)(27,39)(31,37)(35,40)(36,38);(1,3,4,8)(2,5,6,7)(9,10)(11,20)(12,25)(13,18)(14,29)(15,19)(16,24)(17,28)(21,33)(22,36)(23,30)(26,39)(27,32)(31,35)(34,38)(37,40);(2,6)(3,8)(9,11,15,21)(10,16)(12,22,26,35)(13,23,14,24)(17,31)(18,20)(19,30)(25,34)(27,37,28,38)(29,33)(32,36)(39,40);(2,6)(5,7)(11,21)(12,26)(16,30)(17,32)(20,33)(23,24)(25,39)(27,28);(9,12,13,27,15,26,14,28)(10,17,29,39,19,32,18,25)(11,22,23,37,21,35,24,38)(16,31,33,40,30,36,20,34)

Polycyclic presentation:

2^9|b^2=f,c^2=ghi,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,d^c=di,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, 6723648);