#499664

order	512
nilpotency class	4
isoclinism	isoclinic class of #499662
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#23810
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#13334	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1747 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

602 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:

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

Polycyclic presentation:

2^9|a^2=e,d^2=efhi,e^2=hi,f^2=hi,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,d^b=dh,e^b=eh,f^b=fhi,h^b=hi,e^c=eh,f^c=fh,h^c=hi,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, 406860);