#148650

order	512
nilpotency class	4
isoclinism	isoclinic class of #13312
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#469
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

636 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:

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

Polycyclic presentation:

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

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, 55846);