#148144

order	512
nilpotency class	4
isoclinism	isoclinic class of #13313
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#466
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1934
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

682 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=i,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,e^b=eh,f^b=fh,h^b=hi,d^c=di,e^c=eh,f^c=fhi,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, 55340);