#6811499

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806815
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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 ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1939 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)(4,8)(5,7)(9,10,16,19)(11,17,23,20)(12,14,26,15)(13,18,30,35)(21,32,37,36)(22,33,31,24)(25,28,39,29)(27,34,40,38);(1,3,6,2)(4,8,5,7)(9,11)(10,17)(12,24)(13,27)(14,22)(15,31)(16,23)(18,34)(19,20)(21,29)(25,32)(26,33)(28,37)(30,40)(35,38)(36,39);(9,12,15,19,16,26,14,10)(11,20,22,24,23,17,31,33)(13,25,29,35,30,39,28,18)(21,36,27,38,37,32,40,34);(1,4,6,5)(2,7,3,8);(1,5,6,4)(2,7,3,8)(9,13,14,28,16,30,15,29)(10,18,26,39,19,35,12,25)(11,21,31,40,23,37,22,27)(17,32,24,38,20,36,33,34)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=h,d^2=f,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,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6718695);