#501321

order	512
nilpotency class	4
isoclinism	isoclinic class of #501052
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

671 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)(3,7)(4,6)(5,8)(9,10,16,36,34,50,33,49,15,21,14,20,13,19,39,65)(11,17,37,64,25,43,56,69,26,44,24,42,23,41,58,71)(12,27,59,47,29,61,35,18,32,62,30,46,31,51,38,48)(22,52,66,67,28,60,57,40,55,72,53,45,54,70,63,68);(1,3,4,8)(2,5,6,7)(9,11)(10,17)(12,28)(13,23)(14,24)(15,26)(16,37)(18,45)(19,41)(20,42)(21,44)(22,31)(25,34)(27,60)(29,55)(30,63)(32,54)(33,56)(35,53)(36,64)(38,66)(39,58)(40,47)(43,50)(46,68)(48,67)(49,69)(51,52)(57,59)(61,72)(62,70)(65,71);(9,12)(10,18)(11,22)(13,29)(14,35)(15,32)(16,38)(17,40)(19,46)(20,51)(21,48)(23,28)(24,57)(25,54)(26,55)(27,49)(30,33)(31,34)(36,61)(37,63)(39,59)(41,45)(42,70)(43,67)(44,68)(47,50)(52,69)(53,56)(58,66)(60,64)(62,65)(71,72);(2,6)(5,7)(9,13,15,34)(10,19,21,50)(11,23,26,25)(12,29,32,31)(14,33,16,39)(17,41,44,43)(18,46,48,47)(20,49,36,65)(22,28,55,54)(24,56,37,58)(27,61,62,51)(30,38,59,35)(40,45,68,67)(42,69,64,71)(52,60,72,70)(53,63,66,57)

Polycyclic presentation:

2^9|a^2=fi,b^2=e,d^2=i,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, 408517);