#501580

order	512
nilpotency class	4
isoclinism	isoclinic class of #501052
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#41	C₂×D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

493 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:

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

Polycyclic presentation:

2^9|a^2=ef,c^2=i,d^2=ei,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,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, 408776);