#499260

order	512
nilpotency class	4
isoclinism	isoclinic class of #498812
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	C₂²×C₈	C₂⁴
prime radical	#3061	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

798 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:

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

Polycyclic presentation:

2^9|c^2=i,d^2=ei,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=cg,g^b=gi,d^c=di,f^c=fh,g^c=gi,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, 406456);