#6477032

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470402
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

1963 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)(17,18,24,27)(19,22,26,23)(20,29,21,25)(28,31,30,32);(9,11)(10,13)(12,15)(14,16)(17,19,23,18,24,26,22,27)(20,30,31,29,21,28,32,25);(1,3,5,4)(2,6,7,8)(11,15)(13,16)(17,20,24,21)(18,25,27,29)(19,28,26,30)(22,32,23,31);(11,15)(13,16)(17,21,24,20)(18,25,27,29)(19,28,26,30)(22,31,23,32);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(17,22,24,23)(18,19,27,26)(20,31,21,32)(25,30,29,28)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6384228);