#145195

order	512
nilpotency class	4
isoclinism	isoclinic class of #145147
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2034
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#449
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 296 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) = 72
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=dh,d^2=i,e^2=g,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,c^b=cf,d^b=di,e^b=ei,f^b=fh,h^b=hi,d^c=di,e^c=egi,f^c=fh,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52391);