#6822313

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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	#25117	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1359 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)(10,14)(11,12)(17,18,23,26)(19,28,29,24)(20,30,31,22)(21,27,32,25);(9,10)(11,15)(12,16)(13,14)(17,19)(18,24)(20,25)(21,22)(23,29)(26,28)(27,31)(30,32);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(17,20,21,28,23,31,32,24)(18,22,27,19,26,30,25,29);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,14,15)(17,21,23,32)(18,25,26,27)(19,22,29,30)(20,28,31,24);(10,14)(15,16)(17,22,21,19,23,30,32,29)(18,20,27,28,26,31,25,24)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6729509);