#500857

order	512
nilpotency class	4
isoclinism	isoclinic class of #500857
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

964 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)(7,18)(8,21)(9,27)(10,15)(11,17)(12,13)(14,19)(16,28)(20,33)(22,26)(23,25)(24,31)(29,34)(30,32)(35,36);(5,7,19,33,10,22,29,24,11,23,36,32,28,21,9,13)(6,12,27,8,16,30,35,25,17,31,34,26,15,20,14,18);(2,4)(5,6)(7,20)(8,24)(9,14)(10,16)(11,17)(12,22)(13,26)(15,28)(18,33)(19,27)(21,31)(23,30)(25,32)(29,35)(34,36);(5,8,19,30,10,25,29,31,11,26,36,20,28,18,9,12)(6,13,27,7,16,33,35,22,17,24,34,23,15,32,14,21)

Polycyclic presentation:

2^9|b^2=eh,d^2=eh,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,g^b=gi,d^c=di,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, 408053);