#6814821

order	512
nilpotency class	3
isoclinism	isoclinic class of #6814819
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	#60769	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1780 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)(11,15)(12,17)(13,16)(14,19)(18,27)(20,31)(21,29)(22,30)(23,28)(24,33)(25,35)(26,32)(34,38)(36,40)(37,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,24)(13,20)(14,23)(16,31)(17,33)(18,30)(19,28)(21,26)(22,27)(25,36)(29,32)(34,39)(35,40)(37,38);(10,16,19,18)(11,20,23,22)(12,25)(15,28)(17,34,35,32)(21,24,37,36)(26,38)(30,31);(2,6)(5,7);(3,8)(5,7)(9,12,13,26,14,25,27,38)(10,17,16,32,19,35,18,34)(11,21,22,24,23,37,20,36)(15,29,30,33,28,39,31,40)

Polycyclic presentation:

2^9|c^2=ghi,e^2=h,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

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, 6722017);