#262586

order	512
nilpotency class	3
isoclinism	isoclinic class of #246591
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×#550	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1042 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=hi,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=dh,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 169782);