#6989055

order	512
nilpotency class	3
isoclinism	isoclinic class of #6885129
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 ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1608 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6896251);