#152974

order	512
nilpotency class	5
isoclinism	isoclinic class of #13697
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

278 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 94 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)(12,30)(13,34)(14,38)(15,40)(16,43)(17,23)(18,26)(19,31)(20,25)(21,35)(22,41)(27,46)(28,62)(29,63)(32,66)(33,54)(36,50)(37,68)(39,67)(42,47)(44,70)(45,61)(48,57)(49,65)(51,56)(52,55)(53,58)(59,64)(60,69)(71,72);(1,2,3,4)(6,8)(9,11,25,55,15,28,37,60,41,33,65,70,36,59,13,26,17,12,31,56,42,32,39,61,16,29,57,72,35,58,14,27)(10,18,38,64,21,44,48,54,43,69,67,62,47,52,19,24,23,46,34,53,50,71,49,63,22,45,68,66,40,51,20,30);(9,12,31,55,15,32,39,60,41,29,57,70,36,58,14,26,17,11,25,56,42,28,37,61,16,33,65,72,35,59,13,27)(10,18,34,53,21,44,49,63,43,69,68,66,47,52,20,30,23,46,38,64,50,71,48,54,22,45,67,62,40,51,19,24)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=egi,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,c^b=ci

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60170);