#6819710

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	#28225	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1373 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)(4,8)(5,7)(9,10)(11,14)(12,16)(13,15)(17,18,23,26)(19,28,30,22)(20,31,29,24)(21,27,32,25);(1,3,5,4)(2,6,7,8)(9,11,14,10)(12,16,13,15)(17,19)(18,22)(20,32)(21,29)(23,30)(24,25)(26,28)(27,31);(1,4,5,3)(2,6,7,8)(9,12,14,13)(10,15,11,16)(17,20,32,19,23,29,21,30)(18,24,25,22,26,31,27,28);(9,12,14,13)(10,15,11,16)(17,21,23,32)(18,25,26,27)(19,29,30,20)(22,24,28,31);(9,13,14,12)(10,15,11,16)(17,22,21,24,23,28,32,31)(18,19,27,20,26,30,25,29)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gh,d^2=f,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, 6726906);