#6565883

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1960 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,d^2=hi,e^2=h,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6473079);