#6834318

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806575
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	#39133	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#120 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1770 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:

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

Polycyclic presentation:

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

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, 6741514);