#6833133

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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1293 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)(9,10,12,13)(11,14,15,16)(17,18,21,23)(19,26,24,20)(22,25,29,31)(27,28,32,30);(1,2)(3,6)(4,5)(7,8)(18,23)(19,27)(20,28)(22,29)(24,32)(26,30);(10,13)(14,16)(17,19,22,27,21,24,29,32)(18,20,25,30,23,26,31,28);(1,3,4,7)(2,6,5,8)(18,23)(19,24)(25,31)(27,32);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(17,20,29,28,21,26,22,30)(18,24,31,27,23,19,25,32)

Polycyclic presentation:

2^9|a^2=g,c^2=h,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,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, 6740329);