#6695550

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1742 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,d^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6602746);