#6374586

order	512
nilpotency class	3
isoclinism	isoclinic class of #6342659
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#3460	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3297 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,d^2=i,e^2=f,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=cf,d^b=di,e^b=ei,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6281782);