#6964799

order	512
nilpotency class	3
isoclinism	isoclinic class of #6885065
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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1630 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6871995);