#488762

order	512
nilpotency class	4
isoclinism	isoclinic class of #488723
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

776 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 776 conjugacy classes of subgroups — too many to draw. Only its 132 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:

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

Polycyclic presentation:

2^9|d^2=ghi,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dghi,e^a=eg,f^a=fi,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 395958);