#476662

order	512
nilpotency class	3
isoclinism	isoclinic class of #476659
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

973 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=i,c^2=i,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=def,e^b=ei,f^b=fi,g^b=gi,d^c=de,e^c=ei,f^c=fi,g^c=gi,f^d=fi,g^d=gi,h^d=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 383858);