#6821805

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	#31413	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1379 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=h,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, 6729001);