#451786

order	512
nilpotency class	3
isoclinism	isoclinic class of #451785
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16014
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

924 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)(3,15,18,9)(4,10)(5,25)(6,14)(7,13)(8,12)(11,29)(16,31,45,42)(17,46,26,56)(19,33,30,41)(20,49,48,34)(21,51)(22,38)(23,37)(24,36)(27,40)(28,39)(32,47,43,59)(35,55)(44,61,58,64)(50,60,52,63)(53,57)(54,62);(1,3)(2,9)(4,20)(5,26)(6,30)(7,19)(8,18)(10,34)(11,32)(12,15)(13,41)(14,33)(16,22)(17,27)(21,52)(23,45)(24,48)(25,56)(28,47)(29,59)(31,37)(35,61)(36,49)(38,42)(39,43)(40,46)(44,55)(50,53)(51,63)(54,58)(57,60)(62,64);(1,4,8,24)(2,10,12,36)(3,16,18,45)(5,21,27,53)(6,23,7,22)(9,31,15,42)(11,35,39,62)(13,38,14,37)(17,44,26,58)(19,48,30,20)(25,57,40,51)(28,55,29,54)(32,60,43,63)(33,49,41,34)(46,61,56,64)(47,52,59,50);(1,5)(2,11,12,39)(3,17,18,26)(4,21)(6,28)(7,29)(8,27)(9,32)(10,35,36,62)(13,40,14,25)(15,43)(16,44,45,58)(19,47,30,59)(20,50,48,52)(22,54)(23,55)(24,53)(31,60)(33,56)(34,61)(37,57,38,51)(41,46)(42,63)(49,64)

Polycyclic presentation:

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

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, 358982);