#494561

order	512
nilpotency class	4
isoclinism	isoclinic class of #483807
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1284 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=i,e^2=h,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,d^b=di,e^b=eh,f^b=fhi,g^b=gh,h^b=hi,e^c=ehi,f^c=fh,g^c=ghi,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, 401757);