#6830870

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	#42677	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1645 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=f,e^2=gh,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, 6738066);