#394085

order	512
nilpotency class	3
isoclinism	isoclinic class of #393982
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

3888 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:

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

Polycyclic presentation:

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

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, 301281);