#153357

order	512
nilpotency class	5
isoclinism	isoclinic class of #153078
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×D₁₆
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
LCS	G ▷ C₂²×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

215 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 215 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 76
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 76 points:

(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,20)(16,21)(17,22)(18,23)(19,24)(25,46)(26,39)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,49)(34,48)(35,50)(36,51)(37,53)(38,54)(47,59)(52,61)(55,74)(56,71)(57,69)(58,70)(60,67)(62,68)(63,72)(64,73)(65,76)(66,75);(6,9)(10,12)(13,15,16,26)(14,20,21,39)(17,25,30,54)(18,36,31,64)(19,37,32,29)(22,46,43,38)(23,51,44,73)(24,53,45,42)(27,59,57,33)(28,61,58,34)(35,62,63,56)(40,47,69,49)(41,52,70,48)(50,68,72,71)(55,66,60,65)(67,76,74,75);(2,4)(5,7,8,11)(6,10,9,12)(14,21)(15,25)(17,33)(18,34)(19,32)(20,38)(22,47)(23,52)(26,54)(27,28)(29,62)(30,59)(31,61)(35,66)(36,55)(37,56)(39,46)(40,70)(41,69)(42,71)(43,49)(44,48)(50,76)(51,67)(53,68)(57,58)(60,64)(63,65)(72,75)(73,74)

Polycyclic presentation:

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

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, 60553);