#504380

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

Polycyclic presentation:

2^9|b^2=i,d^2=f,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, 411576);