#146778

order	512
nilpotency class	4
isoclinism	isoclinic class of #2192
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1879
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

948 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)(7,20,24,48)(8,27)(9,16)(10,35)(11,17)(12,19)(13,22,40,57)(14,29)(15,33)(18,37)(21,52,55,67)(23,49,58,41)(25,39,26,45)(28,43)(30,42)(31,64)(32,46)(34,47)(36,50)(38,60,66,56)(44,63)(51,62,65,54)(53,68,59,61);(2,4)(5,7,11,25,12,24,37,26)(6,13,17,41,19,40,18,49)(8,21,30,56,32,55,31,60)(9,23,33,57,34,58,10,22)(14,38,43,67,44,66,50,52)(15,45,47,20,35,39,16,48)(27,61,42,65,46,68,64,51)(28,62,63,53,36,54,29,59);(5,8)(6,14)(7,21)(9,29)(10,36)(11,30)(12,32)(13,38)(15,46)(16,42)(17,43)(18,50)(19,44)(20,51)(22,54)(23,59)(24,55)(25,56)(26,60)(27,35)(28,33)(31,37)(34,63)(39,61)(40,66)(41,67)(45,68)(47,64)(48,65)(49,52)(53,58)(57,62)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,d^b=dhi,e^b=eh,f^b=fi,h^b=hi,d^c=dh,e^c=eh,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53974);