#504383

order	512
nilpotency class	4
isoclinism	isoclinic class of #499769
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3061	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

521 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)(11,24)(12,19)(13,34)(14,37)(15,22)(16,43)(17,23)(18,27)(20,35)(21,41)(25,58)(26,45)(28,54)(29,66)(30,60)(31,67)(32,53)(33,50)(36,51)(38,56)(39,52)(40,55)(42,57)(44,61)(46,64)(47,65)(48,68)(49,70)(59,63)(62,72)(69,71);(9,11,17,30)(10,18,23,47)(12,31,13,26)(14,38,42,64)(15,29,16,28)(19,48,34,59)(20,45,53,67)(21,54,55,66)(22,56,43,46)(24,57,60,37)(25,36,62,33)(27,40,65,41)(32,69,35,61)(39,63,70,68)(44,51,71,50)(49,58,52,72);(2,5)(6,8)(9,12,17,13)(10,19,23,34)(11,25,30,62)(14,32,42,35)(15,36,16,33)(18,44,47,71)(20,37,53,57)(21,49,55,52)(22,51,43,50)(24,58,60,72)(26,29,31,28)(27,61,65,69)(38,68,64,63)(39,41,70,40)(45,66,67,54)(46,59,56,48);(1,3,4,7)(2,6,5,8)(9,13,17,12)(10,20,23,53)(11,26,30,31)(14,39,42,70)(15,36,16,33)(18,45,47,67)(19,37,34,57)(21,50,55,51)(22,52,43,49)(24,59,60,48)(25,29,62,28)(27,61,65,69)(32,41,35,40)(38,63,64,68)(44,66,71,54)(46,58,56,72)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=fi,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,e^b=eh,g^b=gi,h^b=hi,g^c=gi,e^d=eh,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411579);