#509427

order	512
nilpotency class	4
isoclinism	isoclinic class of #509097
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

488 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=ei,d^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,h^a=hi,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,h^c=hi,f^d=fh,h^d=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416623);