#152403

order	512
nilpotency class	5
isoclinism	isoclinic class of #13694
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×D₁₆
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

218 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 72
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 72 points:

(2,4)(5,6,7,8)(9,10)(11,21)(12,27)(13,17)(14,33)(15,34)(16,23)(18,22)(19,28)(20,32)(24,41)(25,54)(26,55)(29,56)(30,60)(31,61)(35,50)(36,64)(37,65)(38,46)(39,59)(40,66)(42,45)(43,49)(44,53)(47,58)(48,57)(51,62)(52,71)(63,72)(67,69)(68,70);(9,11,22,45,14,25,47,65,32,53,66,72,59,70,29,50,13,24,46,69,31,52,30,51,15,26,48,64,28,49,12,23)(10,16,27,43,19,36,57,55,34,62,60,71,61,67,38,41,17,35,56,68,39,63,40,44,20,37,58,54,33,42,18,21);(1,2,3,4)(10,17)(16,35)(18,38)(19,39)(20,34)(21,41)(27,56)(33,61)(36,63)(37,62)(40,57)(42,67)(43,68)(44,55)(54,71)(58,60)

Polycyclic presentation:

2^9|a^2=f,b^2=dg,c^2=ei,d^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59599);