#6832530

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806671
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1326 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=fh,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di

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, 6739726);