#501699

order	512
nilpotency class	4
isoclinism	isoclinic class of #501698
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#525	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

603 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408895);