#6557266

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535823
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	C₂×#546	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1753 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:

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

Polycyclic presentation:

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

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, 6464462);