#6422617

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	#34732	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1662 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)(3,5)(4,6)(7,8)(9,10,13,14)(11,16,12,15)(17,18,22,24,21,23,19,26)(20,25,28,31,29,30,27,32);(1,3,4,7)(2,5,6,8)(10,14)(11,12)(18,23)(24,26)(25,30)(31,32);(2,6)(3,7)(9,11,13,12)(10,15,14,16)(17,19,21,22)(18,24,23,26)(20,28,29,27)(25,32,30,31);(2,6)(5,8)(9,12,13,11)(10,16,14,15)(17,20,22,28,21,29,19,27)(18,25,24,31,23,30,26,32);(10,14)(15,16)(20,29)(25,30)(27,28)(31,32)

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=f,d^2=h,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=efi,c^b=cgi,d^b=dfi,d^c=di,e^d=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, 6329813);