#6989684

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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
prime radical	#38026	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1689 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)(9,10,14,13)(11,15,12,16)(17,18,22,26)(19,28,31,24)(20,30,27,23)(21,29,32,25);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,14)(13,16)(17,19)(18,24)(20,29)(21,30)(22,31)(23,32)(25,27)(26,28);(3,8)(5,7)(9,12,14,11)(10,15,13,16)(17,20,23,18,22,27,30,26)(19,29,32,24,31,25,21,28);(2,6)(5,7)(9,10,14,13)(11,15,12,16)(17,21,22,32)(18,25,26,29)(19,30,31,23)(20,24,27,28);(9,13,14,10)(11,15,12,16)(19,31)(21,32)(24,28)(25,29)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6896880);