#513181

order	512
nilpotency class	5
isoclinism	isoclinic class of #513173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

465 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 465 conjugacy classes of subgroups — too many to draw. Only its 122 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)(6,12)(7,17)(8,21)(9,23)(10,27)(13,29)(14,31)(15,34)(16,18)(19,25)(20,24)(22,26)(28,36)(30,32)(33,35);(5,6)(7,18)(8,22)(9,26)(10,14)(11,13)(12,21)(15,23)(16,27)(17,34)(19,29)(20,31)(24,30)(25,35)(28,33)(32,36);(2,4)(6,13)(12,29)(14,16)(15,32)(18,31)(22,35)(26,33)(30,34);(5,7)(6,14)(8,23)(9,24)(10,21)(11,20)(12,30)(13,16)(17,28)(18,22)(19,27)(25,36)(26,33)(29,34)(31,35)

Polycyclic presentation:

2^9|e^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dghi,e^a=eg,g^a=gh,h^a=hi,c^b=ci,d^b=de,e^b=eg,g^b=gh,h^b=hi,e^d=eg,g^d=gh,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×#2263
maximal normal subgroup of	D₄×#2263

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420377);