#502145

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

751 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,e^2=i,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,e^b=ei,f^c=fh,g^c=gh,h^c=hi,e^d=ei

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, 409341);