#497461

order	512
nilpotency class	4
isoclinism	isoclinic class of #33854
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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×#3417	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

945 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)(3,4)(5,6,12,18,13,9,11,17)(7,19,24,46,25,48,23,21)(8,26,30,15,31,45,29,27)(10,34,38,16,39,36,37,33)(14,40,43,55,32,35,42,22)(20,49,53,47,54,50,52,68)(28,44,62,64,63,59,61,57)(41,66,67,65,58,51,56,60);(5,7)(6,14)(8,20)(9,32)(10,35)(11,23)(12,24)(13,25)(15,41)(16,46)(17,42)(18,43)(19,34)(21,33)(22,38)(26,56)(27,58)(28,60)(29,52)(30,53)(31,54)(36,48)(37,55)(39,40)(44,49)(45,67)(47,64)(50,59)(51,61)(57,68)(62,66)(63,65);(2,4)(5,8,13,31)(6,15,9,27)(7,20,25,54)(10,28,39,63)(11,30,12,29)(14,41,32,58)(16,44,33,59)(17,45,18,26)(19,47,48,68)(21,50,46,49)(22,51,55,66)(23,53,24,52)(34,64,36,57)(35,60,40,65)(37,62,38,61)(42,67,43,56);(5,9)(6,13)(7,21)(8,27)(10,36)(11,18)(12,17)(14,40)(15,31)(16,37)(19,24)(20,50)(22,42)(23,48)(25,46)(26,30)(28,57)(29,45)(32,35)(33,38)(34,39)(41,65)(43,55)(44,62)(47,52)(49,54)(51,67)(53,68)(56,66)(58,60)(59,61)(63,64)

Polycyclic presentation:

2^9|a^2=g,c^2=i,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,h^a=hi,d^b=de,e^b=eh,h^b=hi,f^c=fi,g^c=gi,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, 404657);