#509896

order	512
nilpotency class	4
isoclinism	isoclinic class of #509155
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

460 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)(3,4)(5,7)(6,8)(9,10,16,36,35,52,33,18,15,22,13,20,14,21,38,47)(11,17,37,40,26,43,54,46,27,45,25,42,23,44,60,66)(12,28,61,49,30,62,34,19,32,63,29,48,31,51,39,50)(24,55,71,67,53,65,59,41,58,72,56,70,57,69,64,68);(1,3)(2,6)(4,7)(5,8)(9,11)(10,17)(12,24)(13,25)(14,23)(15,27)(16,37)(18,46)(19,41)(20,42)(21,44)(22,45)(26,35)(28,55)(29,56)(30,53)(31,57)(32,58)(33,54)(34,59)(36,40)(38,60)(39,64)(43,52)(47,66)(48,70)(49,67)(50,68)(51,69)(61,71)(62,65)(63,72);(1,4,5,6)(2,3,7,8)(10,18)(11,23)(13,33)(14,35)(16,38)(17,40)(19,28)(20,21)(22,47)(24,53)(25,37)(26,27)(29,34)(30,31)(36,52)(39,61)(41,65)(42,45)(43,66)(44,46)(48,51)(49,62)(50,63)(55,70)(57,58)(59,64)(67,72)(68,69);(2,7)(3,8)(9,12,15,32)(10,19,22,50)(11,24,27,58)(13,34,16,39)(14,30,35,31)(17,41,45,68)(18,28,47,63)(20,51,36,62)(21,48,52,49)(23,53,26,57)(25,59,37,64)(29,38,61,33)(40,65,42,69)(43,67,44,70)(46,55,66,72)(54,56,60,71)

Polycyclic presentation:

2^9|a^2=fi,c^2=e,d^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417092);