#476658

order	512
nilpotency class	3
isoclinism	isoclinic class of #476143
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2819
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17094
prime radical	#23438	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

984 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)(4,21)(5,27,32,44)(6,13)(7,12)(8,14)(9,18)(10,25)(11,31,43,30)(16,46)(17,34,28,41)(19,36)(20,35)(22,54)(23,55,58,38)(24,40)(26,39)(29,42,57,53)(33,49)(37,48)(45,51,62,52)(47,63,59,61)(50,64,56,60);(1,3)(2,9)(4,22)(5,28)(6,19)(7,18)(8,20)(10,37)(11,41)(12,35)(13,15)(14,36)(16,24)(17,32)(21,33)(23,56)(25,49)(26,48)(27,45)(29,59)(30,51)(31,52)(34,43)(38,61)(39,46)(40,54)(42,60)(44,62)(47,57)(50,58)(53,64)(55,63);(1,4)(2,10)(3,16)(5,29)(6,25)(7,24)(8,26)(9,33)(11,42)(12,21)(13,39)(14,40)(15,37)(17,50)(18,48)(19,22)(20,49)(23,30)(27,38)(28,56)(31,58)(32,57)(34,61)(35,54)(36,46)(41,63)(43,53)(44,55)(45,60)(47,51)(52,59)(62,64);(1,5,6,30,8,32,7,31)(2,11,12,27,14,43,13,44)(3,17,18,51,20,28,19,52)(4,23,24,29,26,58,25,57)(9,34,15,45,36,41,35,62)(10,38,39,42,40,55,21,53)(16,47,22,50,49,59,48,56)(33,60,54,61,46,64,37,63)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 383854);