#6816456

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
prime radical	#60935	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1655 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)(9,10,14,18)(11,22,24,16)(12,26,28,38)(13,19,32,25)(15,21,33,20)(17,35,37,30)(23,29,40,31)(27,34,36,39);(1,3,4,8)(2,5,6,7)(9,11,14,24)(10,16,18,22)(12,27,28,36)(13,21,32,20)(15,25,33,19)(17,31,37,29)(23,35,40,30)(26,39,38,34);(2,6)(3,8)(9,12,14,28)(10,17)(11,23)(13,31,32,29)(15,30,33,35)(16,34,22,39)(18,37)(19,27)(20,26)(21,38)(24,40)(25,36);(2,6)(5,7)(10,18)(12,28)(13,32)(16,22)(20,21)(23,40)(27,36)(30,35);(9,13,33,16,14,32,15,22)(10,19,20,11,18,25,21,24)(12,29,35,39,28,31,30,34)(17,36,26,40,37,27,38,23)

Polycyclic presentation:

2^9|a^2=i,b^2=f,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, 6723652);