#471887

order	512
nilpotency class	3
isoclinism	isoclinic class of #456381
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2712
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16808
prime radical	C₂×#2017	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1116 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 379083);