#494537

order	512
nilpotency class	4
isoclinism	isoclinic class of #33531
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

1298 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)(5,6,11,17,34,50,32,48,12,18,10,16,9,15,33,49)(7,19,23,40,31,53,35,13,24,41,22,47,21,51,37,42)(8,25,28,45,26,52,38,14,30,44,27,43,29,54,36,46)(20,39,58,66,56,64,62,63,60,68,57,65,59,67,61,55);(1,2,3,4)(5,7)(6,13)(8,20)(9,31)(10,35)(11,37)(12,24)(14,39)(15,47)(16,51)(17,53)(18,42)(19,48)(21,34)(22,32)(23,33)(25,63)(26,59)(27,62)(28,61)(29,56)(30,60)(36,58)(38,57)(40,50)(41,49)(43,67)(44,55)(45,64)(46,68)(52,66)(54,65);(5,8)(6,14)(7,20)(9,26)(10,36)(11,38)(12,30)(13,39)(15,43)(16,52)(17,54)(18,46)(19,55)(21,56)(22,61)(23,62)(24,60)(25,49)(27,33)(28,32)(29,34)(31,59)(35,58)(37,57)(40,64)(41,63)(42,68)(44,48)(45,50)(47,67)(51,66)(53,65);(5,9,12,34)(6,15,18,50)(7,21,24,31)(8,26,30,29)(10,32,11,33)(13,40,42,47)(14,43,46,45)(16,48,17,49)(19,51,41,53)(20,56,60,59)(22,35,23,37)(25,52,44,54)(27,36,28,38)(39,64,68,67)(55,66,63,65)(57,61,58,62)

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,d^2=i,e^2=hi,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fi,d^b=di,e^b=eh,f^b=fhi,g^b=gh,h^b=hi,e^c=ehi,f^c=fh,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, 401733);