#464653

order	512
nilpotency class	3
isoclinism	isoclinic class of #456501
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1455 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,h^b=hi,d^c=de,e^c=ei,h^c=hi,e^d=ei,h^d=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, 371849);