#501693

order	512
nilpotency class	4
isoclinism	isoclinic class of #500857
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

626 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)(5,6,10,16)(7,18,22,26)(8,23,25,29)(9,27,28,33)(11,17,12,30)(13,14,31,21)(15,24,35,19)(20,36,32,34);(5,7,19,36,12,23,28,14,10,22,24,34,11,29,9,21)(6,13,27,8,17,20,35,26,16,31,33,25,30,32,15,18);(2,4)(5,6)(7,20)(8,21)(9,15)(10,16)(11,30)(12,17)(13,29)(14,25)(18,34)(19,27)(22,32)(23,31)(24,33)(26,36)(28,35);(5,8,24,32,12,26,9,13,10,25,19,20,11,18,28,31)(6,14,33,7,17,34,15,23,16,21,27,22,30,36,35,29)

Polycyclic presentation:

2^9|a^2=f,b^2=eh,d^2=ehi,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, 408889);