#509820

order	512
nilpotency class	4
isoclinism	isoclinic class of #509155
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#24290	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

650 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)(9,10,13,20,14,21,36,53,16,23,38,54,15,22,37,41)(11,17,25,42,39,55,67,51,27,44,59,50,26,43,35,19)(12,28,30,47,31,62,64,49,33,63,65,48,32,52,34,18)(24,46,56,69,60,71,72,68,57,70,66,45,29,61,58,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,29)(13,25)(14,39)(15,26)(16,27)(18,45)(19,41)(20,42)(21,55)(22,43)(23,44)(24,31)(28,61)(30,58)(32,57)(33,60)(34,66)(35,37)(36,67)(38,59)(40,47)(46,62)(48,68)(49,69)(50,54)(51,53)(52,70)(56,64)(63,71)(65,72);(9,12,16,33)(10,18,23,49)(11,24,27,57)(13,34,38,64)(14,32,15,31)(17,40,44,68)(19,46,51,70)(20,52,54,62)(21,48,22,47)(25,58,59,72)(26,60,39,29)(28,53,63,41)(30,36,65,37)(35,56,67,66)(42,61,50,71)(43,69,55,45);(2,6)(3,8)(9,11)(10,19)(12,24)(13,35)(14,26)(15,39)(16,27)(17,41)(18,46)(20,43)(21,50)(22,42)(23,51)(25,37)(28,40)(29,31)(30,58)(32,60)(33,57)(34,56)(36,59)(38,67)(44,53)(45,62)(47,61)(48,71)(49,70)(52,69)(54,55)(63,68)(64,66)(65,72)

Polycyclic presentation:

2^9|a^2=f,c^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, 417016);