#6561641

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535648
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	#2541	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1516 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

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, 6468837);