#6474790

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470884
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1405 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)(3,7)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(18,22)(20,24)(23,29)(25,30)(27,28)(31,32);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(18,23)(19,26)(22,29)(27,28);(1,3,5,4)(2,7,6,8)(11,15)(13,16)(17,18,24,29,21,23,20,22)(19,25,27,32,26,31,28,30);(11,15)(13,16)(17,19,21,26)(18,25,23,31)(20,28,24,27)(22,30,29,32);(1,4,5,3)(2,8,6,7)(10,14)(13,16)(17,20,21,24)(18,22,23,29)(19,27,26,28)(25,32,31,30)

Polycyclic presentation:

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

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, 6381986);