#499472

order	512
nilpotency class	4
isoclinism	isoclinic class of #498860
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

579 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=e,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,g^b=gh,h^b=hi,f^c=fi,g^c=gh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406668);