#147546

order	512
nilpotency class	3
isoclinism	isoclinic class of #2002
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	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1779
prime radical	#3449	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1014 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,c^b=cf,d^b=di,e^b=ei,g^b=gi,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, 54742);