#437717

order	512
nilpotency class	3
isoclinism	isoclinic class of #437716
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15610
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

879 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)(3,13,16,8)(4,18)(5,23,24,53)(6,12)(7,11)(9,21)(10,25,34,54)(14,28,19,37)(15,40)(17,36,26,30)(20,47,48,52)(22,44)(27,35)(29,41)(31,58,43,49)(32,50)(33,55)(38,61)(39,57,46,59)(42,51)(45,63)(56,64)(60,62);(1,3)(2,8)(4,19,21,14)(5,15)(6,26)(7,16)(9,28,18,37)(10,33)(11,13)(12,36)(17,27)(20,38,48,45)(22,39,50,46)(23,51)(24,42)(25,55)(29,34)(30,35)(31,56,43,60)(32,59,44,57)(40,53)(41,54)(47,61,52,63)(49,64,58,62);(1,4)(2,9,11,18)(3,14)(5,20)(6,22)(7,21)(8,28,13,37)(10,31,34,43)(12,32,35,44)(15,38)(16,19)(17,39)(23,52,53,47)(24,48)(25,49)(26,46)(27,50)(29,56,33,60)(30,57,36,59)(40,63,51,61)(41,62)(42,45)(54,58)(55,64);(1,5,7,24)(2,10)(3,15,16,42)(4,20,21,48)(6,25,27,54)(8,29)(9,31)(11,34)(12,23)(13,33)(14,38,19,45)(17,41,26,55)(18,43)(22,49,50,58)(28,56)(30,51)(32,52)(35,53)(36,40)(37,60)(39,62,46,64)(44,47)(57,61)(59,63)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286154, #72286206, #72286267
maximal normal subgroup of	#72286154, #72286206, #72286267

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 344913);