#6767970

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
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

1804 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)(3,7)(4,6)(5,8)(17,18,23,27)(19,24,20,29)(21,26,22,31)(25,28,32,30);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,13)(15,16)(17,19,23,20)(18,24,27,29)(21,30,22,28)(25,31,32,26);(10,13)(14,16)(17,20,23,19)(18,25,27,32)(21,30,22,28)(24,31,29,26);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,21,23,22)(18,26,27,31)(19,28,20,30)(24,32,29,25);(3,8)(5,7)(10,13)(14,16)(19,20)(24,29)(25,32)(28,30)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,d^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,e^c=ei,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, 6675166);