#148410

order	512
nilpotency class	4
isoclinism	isoclinic class of #13389
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2033
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

652 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)(5,6,10,16,12,18,11,37)(7,19,23,53,26,49,25,62)(8,27,30,45,32,47,31,14)(9,33,35,17,36,48,20,15)(13,38,40,24,41,54,52,22)(21,55,57,68,59,42,58,51)(28,39,60,65,56,66,61,67)(29,44,63,50,64,43,34,46);(5,7,20,54,11,25,36,24,12,26,35,38,10,23,9,22)(6,13,33,53,16,40,17,49,18,41,48,62,37,52,15,19)(8,28,34,59,31,61,64,57,32,56,63,21,30,60,29,58)(14,42,46,39,27,51,44,65,45,55,50,66,47,68,43,67);(2,4)(5,8,12,32)(6,14,18,45)(7,21,26,59)(9,34,36,63)(10,31,11,30)(13,39,41,66)(15,46,17,50)(16,47,37,27)(19,51,49,68)(20,29,35,64)(22,60,24,61)(23,58,25,57)(28,54,56,38)(33,43,48,44)(40,67,52,65)(42,62,55,53)

Polycyclic presentation:

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

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, 55606);