#145193

order	512
nilpotency class	4
isoclinism	isoclinic class of #145149
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

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

normal subgroups only

This group has 297 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)(14,36)(15,21)(16,24)(17,30)(18,35)(19,39)(23,54)(25,49)(26,42)(28,64)(29,45)(31,46)(32,43)(33,47)(34,48)(37,70)(38,50)(40,56)(41,65)(44,67)(51,59)(52,57)(53,61)(55,63)(58,71)(60,62)(66,72)(68,69);(9,11,15,26)(10,16,21,43)(12,28,30,66)(13,32,35,24)(14,37,39,65)(17,44,27,62)(18,42,20,22)(19,49,36,53)(23,55,57,31)(25,59,61,34)(29,67,33,60)(38,58,68,56)(40,63,71,46)(41,48,70,51)(45,64,47,72)(50,54,69,52);(2,5)(3,8)(6,7)(9,12,15,30)(10,17)(11,23,26,57)(13,33,35,29)(14,38,39,68)(16,40)(18,47)(19,50)(20,45)(21,27)(22,52)(24,58,32,56)(25,60,61,67)(28,65,66,37)(31,59,55,34)(36,69)(41,64)(42,54)(43,71)(44,49)(46,51)(48,63)(53,62)(70,72)

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,f^a=fi,c^b=cf,d^b=di,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, 52389);