#506384

order	512
nilpotency class	4
isoclinism	isoclinic class of #502469
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

618 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 154 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:

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

Polycyclic presentation:

2^9|b^2=eghi,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^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413580);