#6458744

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437753
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#2619	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2140 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)(9,10,12,14)(11,13,15,16)(17,18,23,26)(19,21,25,22)(20,24,28,31)(27,29,32,30);(9,11)(10,13)(12,15)(14,16)(17,19,22,18,23,25,21,26)(20,27,30,24,28,32,29,31);(2,5)(6,8)(11,15)(13,16)(17,20)(18,24)(19,27)(21,29)(22,30)(23,28)(25,32)(26,31);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(20,28)(24,31)(27,32)(29,30);(17,21,23,22)(18,25,26,19)(20,29,28,30)(24,32,31,27)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6365940);