#6816446

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
prime radical	#3054	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1650 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)(3,7)(4,6)(5,8)(9,10,15,20)(11,16,24,32)(12,18,26,35)(13,17,14,19)(21,30,23,31)(22,29,38,39)(25,34,37,40)(27,33,28,36);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,25)(13,23)(14,21)(15,24)(17,31)(18,34)(19,30)(20,32)(22,28)(26,37)(27,38)(29,36)(33,39)(35,40);(2,6)(3,8)(10,17,20,19)(11,21,24,23)(18,33,35,36)(22,37,38,25);(2,6)(5,7)(12,26)(18,35)(22,38)(25,37)(27,28)(29,39)(33,36)(34,40);(9,12,13,27,15,26,14,28)(10,18,17,33,20,35,19,36)(11,22,21,37,24,38,23,25)(16,29,30,40,32,39,31,34)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gh,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,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, 6723642);