#148103

order	512
nilpotency class	4
isoclinism	isoclinic class of #13307
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1879
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

686 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)(5,6,11,17,12,19,33,18)(7,13,23,40,25,41,24,50)(8,26,30,44,32,62,31,64)(9,16,10,15,34,47,36,48)(14,42,45,63,46,29,51,28)(20,52,54,43,55,68,60,67)(21,56,22,49,35,39,58,38)(27,61,53,37,57,65,59,66);(1,2,3,4)(5,7)(6,13)(8,27)(9,22)(10,35)(11,23)(12,25)(14,43)(15,39)(16,49)(17,40)(18,50)(19,41)(20,29)(21,36)(24,33)(26,61)(28,54)(30,53)(31,59)(32,57)(34,58)(37,44)(38,47)(42,55)(45,68)(46,67)(48,56)(51,52)(60,63)(62,65)(64,66);(5,8)(6,14)(7,20)(9,29)(10,28)(11,30)(12,32)(13,37)(15,44)(16,26)(17,45)(18,51)(19,46)(21,57)(22,59)(23,54)(24,60)(25,55)(27,35)(31,33)(34,42)(36,63)(38,67)(39,68)(40,65)(41,66)(43,49)(47,62)(48,64)(50,61)(52,56)(53,58)

Polycyclic presentation:

2^9|a^2=g,b^2=d,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,e^b=ehi,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, 55299);