#6461840

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437418
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 ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1803 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)(3,8)(4,6)(5,7)(17,18)(19,27)(20,25)(21,31)(22,26)(23,29)(24,30)(28,32);(9,10)(11,15)(12,14)(13,16)(17,19,22,30)(18,23,26,32)(20,29,21,28)(24,31,27,25);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,28)(21,29)(22,30)(23,25)(26,27)(31,32);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,20,22,21)(18,25,26,31)(19,28,30,29)(23,27,32,24);(17,20,22,21)(18,25,26,31)(19,29,30,28)(23,24,32,27)

Polycyclic presentation:

2^9|b^2=i,c^2=gh,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^c=ei

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, 6369036);