#152893

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

332 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 118 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)(14,34)(15,35)(16,24)(18,30)(19,28)(20,33)(22,40)(23,44)(25,55)(26,56)(29,36)(31,62)(32,63)(37,49)(38,67)(39,68)(41,60)(42,48)(43,53)(45,52)(46,51)(47,69)(50,72)(54,64)(57,59)(58,61)(65,71)(66,70);(6,8)(9,11,22,47,14,25,31,54,33,51,59,67,60,71,12,24,13,23,30,53,32,50,61,68,15,26,48,70,28,52,29,49)(10,16,36,65,19,38,42,46,35,64,58,55,63,69,18,21,17,37,27,45,41,66,57,56,20,39,62,72,34,43,40,44);(1,2,3,4)(10,17)(11,23)(18,40)(19,41)(20,35)(24,49)(25,50)(26,51)(27,36)(34,63)(42,57)(47,53)(52,71)(54,68)(58,62)(67,70)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,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,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, 60089);