#499445

order	512
nilpotency class	4
isoclinism	isoclinic class of #498800
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#469
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#2087
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

561 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=eh,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,d^b=di,e^c=ei,f^c=fh,g^c=gi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406641);