#6442392

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437529
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	#24455	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1763 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)(17,18,24,21)(19,25,29,26)(20,22,31,23)(27,32,30,28);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19)(18,25)(20,30)(21,26)(22,28)(23,32)(24,29)(27,31);(10,14)(13,16)(17,20,24,31)(18,22,21,23)(19,26,29,25)(27,32,30,28);(17,21,24,18)(19,27,29,30)(20,22,31,23)(25,32,26,28);(1,3,4,7)(2,6,5,8)(17,22,24,23)(18,20,21,31)(19,28,29,32)(25,30,26,27)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,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,d^c=di,h^c=hi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6349588);