#146160

order	512
nilpotency class	4
isoclinism	isoclinic class of #144073
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1675
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#472
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

194 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=d,c^2=dgh,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,c^b=cf,f^b=fh,h^b=hi,e^c=eg,f^c=fhi,g^c=gi,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53356);