#6477052

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470374
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1927 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)(9,10)(11,13)(12,14)(15,16)(17,18,22,25)(19,26,24,21)(20,23,28,30)(27,32,31,29);(9,11)(10,13)(12,15)(14,16)(17,19,21,18,22,24,26,25)(20,27,29,23,28,31,32,30);(1,3,5,4)(2,6,7,8)(11,15)(13,16)(17,20,22,28)(18,23,25,30)(19,27,24,31)(21,29,26,32);(11,15)(13,16)(20,28)(23,30)(27,31)(29,32);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(20,28)(23,30)(27,31)(29,32)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6384248);