#506295

order	512
nilpotency class	4
isoclinism	isoclinic class of #500681
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3147	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

638 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 638 conjugacy classes of subgroups — too many to draw. Only its 144 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 — 4 generators on 72 points:

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

Polycyclic presentation:

2^9|b^2=g,c^2=i,d^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,d^b=di,f^c=fh,g^c=gi,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, 413491);