#6565785

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1717 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)(23,32)(24,28)(25,30)(26,31)(27,35)(29,34)(33,36);(1,2,3,4)(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,19)(17,20)(21,23,26,24)(22,28,31,32)(25,34,27,33)(29,30,36,35);(7,13)(11,15)(14,19)(18,20)(21,24,26,23)(22,29,31,36)(25,34,27,33)(28,30,32,35);(5,8,10,16)(6,12,9,17)(7,14,13,19)(11,18,15,20)(21,25,26,27)(22,30,31,35)(23,33,24,34)(28,29,32,36);(5,9,10,6)(7,15,13,11)(8,12,16,17)(14,18,19,20)(22,31)(28,32)(29,36)(30,35)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6472981);