#493529

order	512
nilpotency class	4
isoclinism	isoclinic class of #481284
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#578	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3108 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)(5,6,10,15,11,17,31,16)(7,18,21,40,22,45,30,44)(8,13,25,38,23,36,52,39)(9,26,28,14,29,41,57,42)(12,27,33,20,35,49,34,50)(19,43,48,62,46,64,56,65)(24,53,55,37,51,63,68,61)(32,54,59,47,58,67,60,66);(5,7,11,22)(6,12,17,35)(8,19,23,46)(9,27,29,49)(10,30,31,21)(13,32,36,58)(14,40,42,44)(15,34,16,33)(18,26,45,41)(20,28,50,57)(24,54,51,67)(25,56,52,48)(37,62,61,65)(38,60,39,59)(43,53,64,63)(47,55,66,68);(2,4)(5,8)(6,13)(7,19)(9,24)(10,25)(11,23)(12,32)(14,37)(15,38)(16,39)(17,36)(18,43)(20,47)(21,48)(22,46)(26,53)(27,54)(28,55)(29,51)(30,56)(31,52)(33,59)(34,60)(35,58)(40,62)(41,63)(42,61)(44,65)(45,64)(49,67)(50,66)(57,68);(8,23)(13,36)(19,46)(24,51)(25,52)(32,58)(37,61)(38,39)(43,64)(47,66)(48,56)(53,63)(54,67)(55,68)(59,60)(62,65)

Polycyclic presentation:

2^9|a^2=g,b^2=i,e^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=ehi,g^b=gi,h^b=hi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400725);