#6828636

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806623
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	#42586	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1546 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6735832);