#499032

order	512
nilpotency class	4
isoclinism	isoclinic class of #13313
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#466
prime radical	C₂×#3067	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1052 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:

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

Polycyclic presentation:

2^9|c^2=e,d^2=i,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,f^c=fhi,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, 406228);