#6818189

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×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1515 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=ghi,d^2=i,e^2=fh,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, 6725385);