#494521

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)(5,6,12,19,38,16,37,18,13,21,10,17,9,20,11,40)(7,22,27,43,24,44,26,14,29,45,25,53,28,51,41,46)(8,30,34,50,31,48,39,47,36,52,32,15,35,49,33,54)(23,56,60,55,63,68,59,64,61,67,58,65,57,42,62,66);(1,2,3,4)(5,7)(6,14)(8,23)(9,24)(10,26)(11,27)(12,41)(13,29)(15,42)(16,43)(17,51)(18,22)(19,44)(20,53)(21,46)(25,37)(28,38)(30,64)(31,57)(32,59)(33,60)(34,62)(35,63)(36,61)(39,58)(40,45)(47,56)(48,55)(49,65)(50,68)(52,66)(54,67);(5,8)(6,15)(7,23)(9,31)(10,39)(11,34)(12,33)(13,36)(14,42)(16,47)(17,30)(18,48)(19,52)(20,54)(21,50)(22,55)(24,57)(25,59)(26,58)(27,62)(28,63)(29,61)(32,37)(35,38)(40,49)(41,60)(43,56)(44,66)(45,65)(46,68)(51,64)(53,67);(5,9,13,38)(6,16,21,20)(7,24,29,28)(8,31,36,35)(10,37,12,11)(14,43,46,53)(15,47,50,54)(17,40,19,18)(22,51,45,44)(23,57,61,63)(25,41,27,26)(30,49,52,48)(32,33,34,39)(42,56,68,67)(55,64,65,66)(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,f^a=fi,e^b=eh,f^b=fh,g^b=gh,h^b=hi,e^c=eh,f^c=fhi,g^c=gh,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, 401717);