#6836305

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806551
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	#39856	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1483 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:

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

Polycyclic presentation:

2^9|c^2=ghi,d^2=fi,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,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6743501);