#148128

order	512
nilpotency class	4
isoclinism	isoclinic class of #13312
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#469
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1999
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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=i,e^2=hi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fh,h^a=hi,c^b=cf,e^b=eh,f^b=fhi,h^b=hi,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, 55324);