#6442124

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437521
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	#24425	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1750 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,28)(20,25)(21,32)(22,27)(23,26)(24,30)(29,31);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,18)(19,29)(20,24)(21,26)(22,27)(23,32)(25,30)(28,31);(10,14)(13,16)(17,19)(18,24)(20,21)(22,30)(23,31)(25,26)(27,28)(29,32);(17,20,22,31)(18,25,27,29)(19,21,30,23)(24,26,28,32);(1,3,4,7)(2,6,5,8)(17,21,22,23)(18,26,27,32)(19,20,30,31)(24,25,28,29)

Polycyclic presentation:

2^9|b^2=gi,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,h^c=hi,h^d=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, 6349320);