#504668

order	512
nilpotency class	4
isoclinism	isoclinic class of #500733
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	C₂×#41	C₂×D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

508 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)(11,24)(12,30)(13,37)(14,22)(15,21)(16,23)(17,27)(18,33)(19,38)(20,40)(25,57)(26,53)(28,46)(29,54)(31,68)(32,51)(34,48)(35,50)(36,55)(39,52)(41,70)(42,67)(43,61)(44,64)(45,63)(47,69)(49,58)(56,60)(59,72)(62,66)(65,71);(1,3,5,4)(2,6,7,8)(9,11,14,27,16,29,40,28)(10,17,20,24,23,46,22,54)(12,31,33,62,36,69,35,59)(13,26,38,63,39,64,15,41)(18,47,30,66,50,68,55,72)(19,44,37,45,21,53,52,70)(25,58,60,32,61,42,65,34)(43,48,56,67,57,51,71,49);(1,4,5,3)(2,6,7,8)(9,12,14,33,16,36,40,35)(10,18,20,30,23,50,22,55)(11,25,27,60,29,61,28,65)(13,34,38,58,39,32,15,42)(17,43,24,56,46,57,54,71)(19,51,37,49,21,48,52,67)(26,62,63,69,64,59,41,31)(44,68,45,72,53,47,70,66);(2,7)(6,8)(9,13,14,38,16,39,40,15)(10,19,20,37,23,21,22,52)(11,26,27,63,29,64,28,41)(12,32,33,42,36,34,35,58)(17,44,24,45,46,53,54,70)(18,48,30,67,50,51,55,49)(25,59,60,31,61,62,65,69)(43,47,56,66,57,68,71,72)

Polycyclic presentation:

2^9|b^2=e,c^2=e,d^2=f,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gh,h^b=hi,d^c=di,g^c=gh,h^c=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, 411864);