#148292

order	512
nilpotency class	4
isoclinism	isoclinic class of #13392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#482
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#2034
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

658 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,h^a=hi,c^b=cf,f^b=fi,g^b=gi,f^c=fi,g^c=gi

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, 55488);