#148558

order	512
nilpotency class	4
isoclinism	isoclinic class of #13402
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2034
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

634 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)(5,6,12,18,13,20,38,19)(7,21,26,51,27,53,40,42)(8,28,32,47,34,48,33,65)(9,17,10,16,35,52,11,39)(14,25,43,23,45,24,44,36)(15,30,49,31,50,64,54,37)(22,41,57,67,59,61,58,68)(29,62,63,66,60,55,56,46);(5,7,13,27)(6,14,20,45)(8,29,34,60)(9,25,35,24)(10,36,11,23)(12,40,38,26)(15,46,50,66)(16,51,39,42)(17,53,52,21)(18,44,19,43)(22,37,59,31)(28,61,48,41)(30,57,64,58)(32,56,33,63)(47,67,65,68)(49,55,54,62);(2,4)(5,8,9,30,12,33,10,37,13,34,35,64,38,32,11,31)(6,15,16,47,19,49,17,48,20,50,39,65,18,54,52,28)(7,22,23,56,26,58,24,60,27,59,36,63,40,57,25,29)(14,41,21,55,44,67,42,46,45,61,53,62,43,68,51,66)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=d,d^2=h,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fh,h^a=hi,c^b=cf,d^b=dhi,e^b=eh,f^b=fh,g^b=gi,h^b=hi,e^c=ei,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, 55754);