#476443

order	512
nilpotency class	3
isoclinism	isoclinic class of #476443
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2829
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17089
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

812 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=def,g^b=gi,h^b=hi,d^c=de,e^c=ei,h^c=hi,e^d=ei,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, 383639);