#394112

order	512
nilpotency class	3
isoclinism	isoclinic class of #394046
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62859	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3885 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,f^b=fi,e^c=ei,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, 301308);