#498171

order	512
nilpotency class	4
isoclinism	isoclinic class of #33854
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#578	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

727 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=i,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,h^a=hi,d^b=deh,e^b=eh,h^b=hi,d^c=di,f^c=fi,e^d=eh,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405367);