#146069

order	512
nilpotency class	4
isoclinism	isoclinic class of #146048
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2073
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#463
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

222 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 70 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:

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

Polycyclic presentation:

2^9|a^2=egh,b^2=i,c^2=d,e^2=gi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,f^b=fh,h^b=hi,e^c=eg,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, 53265);