#501698

order	512
nilpotency class	4
isoclinism	isoclinic class of #501698
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

629 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=eh,d^2=eh,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,d^b=di,f^c=fi

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, 408894);