#6450242

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1774 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)(9,10,12,13)(11,14,15,16)(18,24)(19,26)(25,31)(28,29);(2,5)(6,8)(17,18)(19,27)(20,29)(21,25)(22,24)(23,31)(26,32)(28,30);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,19,22,26)(18,20,24,30)(21,29,23,28)(25,32,31,27);(9,11,12,15)(10,14,13,16)(17,20,22,30)(18,19,24,26)(21,27,23,32)(25,28,31,29);(17,21,22,23)(18,25,24,31)(19,28,26,29)(20,27,30,32)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6357438);