#419921

order	512
nilpotency class	3
isoclinism	isoclinic class of #419919
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1524 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 327117);