#6565846

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1729 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

(1,2)(3,4)(5,6,10,9)(7,11,13,15)(8,12,16,17)(14,18,19,20)(21,22,26,30)(23,31,32,35)(24,33,27,28)(25,34,36,29);(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,19)(17,20)(21,23,25,33,26,32,36,28)(22,27,29,31,30,24,34,35);(2,4)(7,13)(11,15)(14,19)(18,20)(21,24)(22,28)(23,29)(25,35)(26,27)(30,33)(31,36)(32,34);(5,8,10,16)(6,12,9,17)(7,14,13,19)(11,18,15,20)(23,32)(24,27)(28,33)(31,35);(5,9,10,6)(7,15,13,11)(8,12,16,17)(14,18,19,20)

Polycyclic presentation:

2^9|a^2=h,b^2=f,d^2=hi,e^2=hi,f^2=i,b^a=bf,c^a=cg,f^a=fi,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, 6473042);