#427884

order	512
nilpotency class	3
isoclinism	isoclinic class of #427883
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15375
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1183 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:

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

Polycyclic presentation:

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

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, 335080);