#6484589

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
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	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1742 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18,22,23,21,26,20,25)(19,24,29,30,27,32,28,31);(2,5)(6,8)(10,14)(11,12)(18,23)(19,27)(20,22)(24,31)(25,26)(30,32);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,14,16)(18,23)(20,22)(24,30)(25,26)(28,29)(31,32);(9,12,13,11)(10,16,14,15)(17,19,21,27)(18,24,26,32)(20,28,22,29)(23,30,25,31);(10,14)(15,16)(19,27)(24,32)(28,29)(30,31)

Polycyclic presentation:

2^9|a^2=gi,c^2=fgi,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dhi,f^b=fi,g^b=gi,f^c=fi,g^c=gi,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, 6391785);