#6841172

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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1303 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)(11,15)(12,16)(13,14)(17,18,23,22)(19,24,26,21)(20,25,29,31)(27,28,32,30);(1,2,4,5)(3,6,7,8)(17,19,23,26)(18,24,22,21)(20,28,29,30)(25,32,31,27);(10,14)(15,16)(17,20,18,25,23,29,22,31)(19,27,21,30,26,32,24,28);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,14,16)(20,29)(25,31)(27,32)(28,30);(2,5)(6,8)(9,12,13,11)(10,16,14,15)(17,21,22,19,23,24,18,26)(20,30,31,27,29,28,25,32)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=hi,d^2=fg,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, 6748368);