#6536089

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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	#23860	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2037 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)(3,8)(4,6)(5,7)(17,18,22,26)(19,23,28,32)(20,25,21,24)(27,31,29,30);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,28)(18,23,26,32)(20,27,21,29)(24,30,25,31);(10,13)(14,16)(18,24)(20,21)(23,30)(25,26)(27,29)(31,32);(1,4,5,3)(2,6,7,8)(9,11)(10,14)(12,15)(13,16)(17,20,22,21)(18,25,26,24)(19,27,28,29)(23,31,32,30);(11,15)(14,16)(19,28)(23,32)(27,29)(30,31)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,d^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ei,d^c=di,g^c=gi,e^d=eh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6443285);