#502134

order	512
nilpotency class	4
isoclinism	isoclinic class of #502133
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

744 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409330);