#148506

order	512
nilpotency class	4
isoclinism	isoclinic class of #136735
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1879
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

646 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)(7,20)(8,27,32,57)(9,17)(10,36)(11,16)(12,38)(13,19)(14,22)(15,30,46,31)(18,34)(21,40,54,64)(23,42)(24,49)(25,56)(26,51)(28,50,53,67)(29,47,62,63)(33,45,61,58)(35,48)(37,43)(39,41)(44,60,66,52)(55,68,59,65);(1,2)(3,4)(5,7,13,26)(6,14,19,43)(8,28,32,53)(9,24,35,23)(10,37,11,22)(12,39,34,25)(15,44,46,66)(16,41,36,56)(17,20,48,51)(18,49,38,42)(21,30,54,31)(27,40,57,64)(29,55,62,59)(33,52,61,60)(45,65,58,68)(47,50,63,67);(2,4)(5,8,9,29,34,61,10,30,13,32,35,62,12,33,11,31)(6,15,16,45,38,63,48,57,19,46,36,58,18,47,17,27)(7,21,22,52,39,59,23,53,26,54,37,60,25,55,24,28)(14,40,20,50,49,68,56,66,43,64,51,67,42,65,41,44)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55702);