#148412

order	512
nilpotency class	4
isoclinism	isoclinic class of #13334
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2011
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

668 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)(7,19)(8,14)(9,30)(10,15)(11,34)(12,18)(13,21)(16,32)(17,31)(20,40)(22,51)(23,50)(24,38)(25,60)(26,45)(27,41)(28,48)(29,44)(33,46)(35,43)(36,61)(37,57)(39,58)(42,63)(47,64)(49,54)(52,56)(53,68)(55,67)(59,65)(62,66);(5,7,10,23,11,24,32,57,12,22,33,58,31,56,9,21)(6,13,30,52,17,39,46,51,18,37,16,38,34,50,15,19)(8,25,27,55,28,62,63,54,29,61,64,59,35,53,26,20)(14,40,45,68,43,65,47,36,44,49,42,66,48,67,41,60);(5,8,12,29)(6,14,18,44)(7,20,22,54)(9,27,32,64)(10,26,33,63)(11,35,31,28)(13,36,37,60)(15,45,46,42)(16,47,30,41)(17,48,34,43)(19,49,51,40)(21,25,57,61)(23,53,58,62)(24,59,56,55)(38,67,52,65)(39,68,50,66)

Polycyclic presentation:

2^9|a^2=g,b^2=dh,c^2=i,d^2=hi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,h^a=hi,c^b=cf,e^b=ei,d^c=dh,f^c=fhi,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, 55608);