#6431637

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#52294	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1538 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=g,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cgi,d^b=dfi,e^c=ei

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, 6338833);