#509849

order	512
nilpotency class	4
isoclinism	isoclinic class of #509155
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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

836 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 136 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

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

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, 417045);