#413893

order	512
nilpotency class	3
isoclinism	isoclinic class of #401009
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1345 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 321089);