#507193

order	512
nilpotency class	4
isoclinism	isoclinic class of #502469
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	#3061	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

553 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)(3,8)(4,6)(5,7)(9,10)(11,21)(12,32)(13,39)(14,42)(15,44)(16,24)(17,46)(18,25)(19,27)(20,36)(22,40)(23,43)(26,59)(28,49)(29,57)(30,48)(31,58)(33,68)(34,65)(35,54)(37,66)(38,67)(41,50)(45,53)(47,63)(51,64)(52,62)(55,69)(56,60)(61,71)(70,72);(1,3,5,4)(2,6,7,8)(9,11,14,27,16,29,15,28,18,31,43,41,17,30,40,13)(10,19,44,58,46,39,42,57,25,50,22,21,24,49,23,48)(12,33,69,62,45,26,60,70,38,64,35,34,37,63,36,61)(20,51,53,71,54,59,32,65,56,68,66,72,55,47,67,52);(1,4,5,3)(2,6,7,8)(9,12,14,35,16,45,15,36,18,38,43,69,17,37,40,60)(10,20,22,53,46,54,23,32,25,56,44,66,24,55,42,67)(11,26,27,61,29,64,28,62,31,63,41,70,30,33,13,34)(19,47,21,52,39,51,48,71,50,59,58,65,49,68,57,72);(2,7)(6,8)(9,13,40,30,17,41,43,31,18,28,15,29,16,27,14,11)(10,21,42,19,24,57,44,49,25,58,23,50,46,48,22,39)(12,34,60,33,37,70,69,63,38,62,36,64,45,61,35,26)(20,52,67,47,55,72,66,68,56,65,32,59,54,71,53,51)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414389);