#506385

order	512
nilpotency class	4
isoclinism	isoclinic class of #502472
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

612 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)(7,20)(8,15)(9,14)(10,31)(11,18)(12,23)(13,19)(16,21)(17,27)(22,26)(24,32)(25,28)(29,35)(30,34)(33,36);(5,7,10,9,11,21,32,28,13,15,33,30,12,22,27,29)(6,14,24,8,23,35,31,16,19,34,17,20,18,25,36,26);(2,4)(5,8)(6,15)(7,19)(9,24)(10,25)(11,16)(12,26)(13,20)(14,32)(17,30)(18,21)(22,23)(27,34)(28,31)(29,36)(33,35);(5,9,27,7,12,29,33,22,13,30,32,15,11,28,10,21)(6,16,31,25,18,8,24,34,19,26,36,35,23,20,17,14)

Polycyclic presentation:

2^9|b^2=egh,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,d^b=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, 413581);