#502140

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=i,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, 409336);