#6798249

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682532
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1857 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)(4,8)(5,7)(10,13)(11,15)(17,18)(19,27)(20,30)(21,26)(22,25)(23,31)(24,28)(29,32);(1,3,6,2)(4,8,5,7)(9,10,12,13)(11,14,15,16)(17,19,22,28)(18,24,25,27)(20,31,32,26)(21,29,23,30);(10,13)(14,16)(17,20,23,27,22,32,21,24)(18,19,26,30,25,28,31,29);(1,4,6,5)(2,7,3,8)(9,11,12,15)(10,14,13,16)(18,25)(19,28)(26,31)(29,30);(1,5,6,4)(2,7,3,8)(17,21,22,23)(18,26,25,31)(19,29,28,30)(20,27,32,24)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6705445);