#6476302

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470003
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#23923	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1776 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,21)(19,26,29,27)(20,30,25,22)(24,32,31,28);(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,26)(21,31)(22,32)(23,29)(25,27)(28,30);(2,5)(6,8)(11,15)(13,16)(17,20)(18,22)(19,27)(21,30)(23,25)(24,28)(26,29)(31,32);(11,15)(13,16)(17,21,23,18)(19,24,29,31)(20,30,25,22)(26,32,27,28);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(18,21)(20,25)(24,31)(26,27)

Polycyclic presentation:

2^9|a^2=i,d^2=i,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^c=ei,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, 6383498);