#152970

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

282 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 98 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)(11,24)(12,31)(13,36)(14,38)(15,41)(16,23)(17,26)(18,27)(19,34)(20,25)(21,32)(22,39)(28,62)(29,65)(30,42)(33,50)(35,69)(37,67)(40,47)(43,70)(44,61)(45,71)(46,60)(48,56)(49,68)(51,59)(52,55)(53,57)(54,63)(58,66)(64,72);(1,2,3,4)(6,8)(9,11,25,55,14,28,35,60,39,63,68,70,33,58,12,26,16,30,34,59,40,64,37,61,15,29,56,71,32,57,13,27)(10,17,36,66,21,43,48,54,41,46,67,62,47,52,19,24,23,18,31,53,50,45,49,65,22,44,69,72,38,51,20,42);(9,11,25,55,14,28,35,60,39,63,68,70,33,58,12,26,16,30,34,59,40,64,37,61,15,29,56,71,32,57,13,27)(10,18,36,53,21,45,48,65,41,44,67,72,47,51,19,42,23,17,31,66,50,43,49,54,22,46,69,62,38,52,20,24)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=eg,d^2=gh,e^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,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, 60166);