#394056

order	512
nilpotency class	3
isoclinism	isoclinic class of #394055
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3828 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)(8,24)(9,29)(10,17)(11,16)(12,18)(13,22)(14,27)(15,30)(20,35)(21,36)(23,37)(25,39)(26,40)(28,41)(31,43)(32,38)(33,42)(34,44)(45,50)(46,65)(47,61)(48,52)(49,58)(51,62)(53,56)(54,60)(55,59)(57,64)(63,66)(67,68);(2,4)(5,7)(6,13)(8,25)(9,21)(10,32)(11,22)(12,23)(14,39)(15,36)(16,37)(17,19)(18,38)(20,27)(24,52)(26,54)(28,47)(29,58)(30,50)(31,51)(33,48)(34,49)(35,41)(40,63)(42,61)(43,45)(44,62)(46,55)(53,68)(56,66)(57,67)(59,60)(64,65);(5,8)(6,14)(7,20)(9,26)(10,33)(11,27)(12,28)(13,35)(15,40)(16,41)(17,24)(18,42)(19,39)(21,46)(22,47)(23,48)(25,32)(29,59)(30,56)(31,57)(34,55)(36,60)(37,61)(38,52)(43,53)(44,64)(45,63)(49,67)(50,54)(51,66)(58,65)(62,68);(5,9,12,31)(6,15,18,44)(7,21,23,51)(8,26,28,57)(10,30,11,34)(13,36,38,62)(14,40,42,64)(16,29,17,43)(19,45,37,58)(20,46,48,66)(22,49,32,50)(24,53,41,59)(25,54,47,67)(27,55,33,56)(35,60,52,68)(39,63,61,65)

Polycyclic presentation:

2^9|d^2=i,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi,h^d=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, 301252);