#507199

order	512
nilpotency class	4
isoclinism	isoclinic class of #502472
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

551 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)(12,33)(13,40)(14,43)(15,44)(16,24)(17,46)(18,25)(19,28)(20,37)(22,41)(23,26)(27,60)(29,49)(30,57)(31,48)(32,58)(34,68)(35,65)(36,54)(38,66)(39,67)(42,50)(45,53)(47,63)(51,64)(52,62)(55,59)(56,69)(61,71)(70,72);(1,3,5,4)(2,6,7,8)(9,11,26,42,16,30,41,13,18,32,14,28,17,31,15,29)(10,19,22,21,46,40,23,48,25,50,44,58,24,49,43,57)(12,34,36,35,45,27,37,61,39,64,59,62,38,63,69,70)(20,51,66,72,54,60,67,52,56,68,53,71,55,47,33,65);(1,4,5,3)(2,6,7,8)(9,12,14,36,16,45,15,37,18,39,26,59,17,38,41,69)(10,20,22,53,46,54,23,33,25,56,44,66,24,55,43,67)(11,27,28,61,30,64,29,62,32,63,42,70,31,34,13,35)(19,47,21,52,40,51,48,71,50,60,58,65,49,68,57,72);(2,7)(6,8)(9,13,41,31,17,42,26,32,18,29,15,30,16,28,14,11)(10,21,43,19,24,57,44,49,25,58,23,50,46,48,22,40)(12,35,69,34,38,70,59,63,39,62,37,64,45,61,36,27)(20,52,67,47,55,72,66,68,56,65,33,60,54,71,53,51)

Polycyclic presentation:

2^9|b^2=eghi,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,d^b=di,g^c=gi

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, 414395);