#147530

order	512
nilpotency class	3
isoclinism	isoclinic class of #2021
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#120
commutator	C₂⁴	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1779
prime radical	#3449	C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1020 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=g,g^2=h,h^2=i,b^a=bd,c^a=ce,c^b=cf,e^b=ei,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54726);