#506381

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

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,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, 413577);