#6827010

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	#28485	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1537 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)(11,14)(12,16)(13,15)(17,18,23,28)(19,29,30,25)(20,22,27,32)(21,31,24,26);(9,11,14,10)(12,16,13,15)(17,19)(18,25)(20,24)(21,27)(22,31)(23,30)(26,32)(28,29);(2,5)(6,8)(9,12,14,13)(10,15,11,16)(17,20,21,30,23,27,24,19)(18,22,31,25,28,32,26,29);(1,3,4,7)(2,6,5,8)(9,12,14,13)(10,15,11,16)(17,21,23,24)(18,26,28,31)(19,27,30,20)(22,25,32,29);(9,13,14,12)(10,15,11,16)(17,22,24,29,23,32,21,25)(18,27,26,30,28,20,31,19)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=fh,d^2=gi,e^2=fhi,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, 6734206);