#6477023

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470428
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

1929 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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=dhi,e^d=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, 6384219);