#6763497

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1706 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 6670693);