#6841212

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806903
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 ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1316 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)(17,18,24,30)(19,25,31,32)(20,26,22,28)(21,27,23,29);(1,2,4,5)(3,6,7,8)(17,19,24,31)(18,25,30,32)(20,21,22,23)(26,27,28,29);(10,14)(15,16)(17,20,23,19,24,22,21,31)(18,26,29,25,30,28,27,32);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,14,16)(17,21,24,23)(18,27,30,29)(19,22,31,20)(25,28,32,26);(2,5)(6,8)(9,12,13,11)(10,16,14,15)(17,22,23,31,24,20,21,19)(18,28,29,32,30,26,27,25)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=h,d^2=fgi,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^d=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, 6748408);