#494529

order	512
nilpotency class	4
isoclinism	isoclinic class of #33539
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
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

1290 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)(5,6,12,19,38,16,37,50,13,21,10,17,9,20,11,18)(7,22,27,42,24,43,39,14,29,44,25,53,28,51,26,45)(8,30,34,49,31,47,33,46,36,52,32,15,35,48,40,54)(23,56,60,65,63,68,59,64,61,67,58,55,57,41,62,66);(1,2,3,4)(5,7)(6,14)(8,23)(9,24)(10,39)(11,27)(12,26)(13,29)(15,41)(16,42)(17,51)(18,44)(19,43)(20,53)(21,45)(22,50)(25,37)(28,38)(30,64)(31,57)(32,59)(33,58)(34,62)(35,63)(36,61)(40,60)(46,56)(47,65)(48,55)(49,68)(52,66)(54,67);(5,8)(6,15)(7,23)(9,31)(10,40)(11,32)(12,33)(13,36)(14,41)(16,46)(17,52)(18,47)(19,30)(20,54)(21,49)(22,55)(24,57)(25,62)(26,58)(27,59)(28,63)(29,61)(34,37)(35,38)(39,60)(42,56)(43,64)(44,65)(45,68)(48,50)(51,66)(53,67);(5,9,13,38)(6,16,21,20)(7,24,29,28)(8,31,36,35)(10,37,12,11)(14,42,45,53)(15,46,49,54)(17,18,19,50)(22,51,44,43)(23,57,61,63)(25,26,27,39)(30,48,52,47)(32,40,34,33)(41,56,68,67)(55,66,65,64)(58,59,60,62)

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,d^2=i,e^2=hi,f^2=h,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,e^b=eh,f^b=fhi,g^b=gh,h^b=hi,e^c=ehi,f^c=fhi,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, 401725);