#489382

order	512
nilpotency class	4
isoclinism	isoclinic class of #489379
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#62439	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

747 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 72 points:

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

Polycyclic presentation:

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

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, 396578);