#435784

order	512
nilpotency class	3
isoclinism	isoclinic class of #435783
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15538
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

977 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 342980);