#419828

order	512
nilpotency class	3
isoclinism	isoclinic class of #419810
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
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

1537 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)(7,18,21,13)(8,14)(9,29,30,61)(10,33)(11,17)(12,16)(15,32,42,31)(19,34,47,45)(20,49)(22,36,23,43)(24,53,48,37)(25,55,56,60)(26,44)(27,40)(28,39)(35,51)(38,58,65,57)(41,50)(46,63)(52,59)(54,62)(64,68)(66,67);(1,2)(3,4)(5,7)(6,13)(8,24)(9,20)(10,23)(11,22)(12,21)(14,37)(15,41)(16,18)(17,43)(19,26)(25,54)(27,47)(28,48)(29,59)(30,52)(31,51)(32,50)(33,36)(34,40)(35,42)(38,64)(39,53)(44,45)(46,57)(49,61)(55,66)(56,67)(58,68)(60,62)(63,65);(2,4)(5,8,12,28)(6,14,16,39)(7,19,21,47)(9,25,30,56)(10,27,11,26)(13,34,18,45)(15,38,42,65)(17,44,33,40)(20,46,52,68)(22,48,23,24)(29,60,61,55)(31,58,32,57)(35,62,41,66)(36,53,43,37)(49,63,59,64)(50,67,51,54);(5,9,12,30)(6,15)(7,20,21,52)(8,25,28,56)(10,31,11,32)(13,35)(14,38)(16,42)(17,29)(18,41)(19,46,47,68)(22,50,23,51)(24,54,48,67)(26,57,27,58)(33,61)(34,62)(36,59)(37,63)(39,65)(40,55)(43,49)(44,60)(45,66)(53,64)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=ei,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,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, 327024);