#509853

order	512
nilpotency class	4
isoclinism	isoclinic class of #509163
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

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

normal subgroups only

This group has 834 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:

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

Polycyclic presentation:

2^9|a^2=efi,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=defi,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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417049);