#504256

order	512
nilpotency class	4
isoclinism	isoclinic class of #500730
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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

511 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=f,c^2=f,d^2=e,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,g^c=gh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411452);