#6461833

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437415
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1762 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)(19,25)(20,29)(21,26)(22,31)(23,27)(24,28)(30,32);(9,10)(11,15)(12,14)(13,16)(17,19,23,20)(18,24,27,32)(21,28,22,30)(25,26,29,31);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20,23,19)(18,25,27,29)(21,28,22,30)(24,26,32,31);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,21,23,22)(18,26,27,31)(19,28,20,30)(24,29,32,25);(18,27)(24,32)(25,29)(26,31)

Polycyclic presentation:

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

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, 6369029);