#419797

order	512
nilpotency class	3
isoclinism	isoclinic class of #419794
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1536 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)(5,6,11,17,12,18,10,16)(7,19,23,13,24,36,22,37)(8,25,27,40,28,45,33,14)(9,15,31,43,32,44,30,42)(20,38,48,61,49,34,52,46)(21,41,50,35,51,60,29,57)(26,53,55,64,56,66,58,39)(47,62,54,63,67,59,68,65);(5,7,12,24)(6,13,18,37)(8,20,28,49)(9,29,32,50)(10,23,11,22)(14,38,40,34)(15,41,44,60)(16,36,17,19)(21,31,51,30)(25,46,45,61)(26,54,56,68)(27,52,33,48)(35,43,57,42)(39,63,64,65)(47,58,67,55)(53,62,66,59);(5,8)(6,14)(7,20)(9,26)(10,27)(11,33)(12,28)(13,34)(15,39)(16,25)(17,45)(18,40)(19,46)(21,47)(22,48)(23,52)(24,49)(29,54)(30,55)(31,58)(32,56)(35,59)(36,61)(37,38)(41,65)(42,53)(43,66)(44,64)(50,68)(51,67)(57,62)(60,63);(2,4)(5,9,11,31,12,32,10,30)(6,15,17,43,18,44,16,42)(7,21,23,50,24,51,22,29)(8,26,33,58,28,56,27,55)(13,35,36,60,37,57,19,41)(14,39,45,66,40,64,25,53)(20,47,52,68,49,67,48,54)(34,59,61,63,38,62,46,65)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=ei,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,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, 326993);