#268541

order	512
nilpotency class	3
isoclinism	isoclinic class of #268491
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1004 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,d^2=h,e^2=i,f^2=hi,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,e^c=ehi,f^c=fhi,e^d=ei,f^d=fhi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 175737);