#509343

order	512
nilpotency class	4
isoclinism	isoclinic class of #509340
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#30593
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#13310	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

532 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)(3,10,21,41)(4,22,5,26)(7,30,29,47)(8,15,9,14)(11,27,42,24)(13,34,44,35)(16,49,51,64)(17,52,23,58)(18,54,31,60)(19,40,20,39)(25,59,28,46)(32,48,33,43)(36,61,62,50)(37,55,63,53)(38,56,45,57);(1,3,9,20,6,21,8,19)(2,10,15,40,12,41,14,39)(4,23,25,51,5,17,28,16)(7,31,35,56,29,18,34,57)(11,37,43,62,42,63,48,36)(13,45,30,60,44,38,47,54)(22,58,46,49,26,52,59,64)(24,53,32,50,27,55,33,61);(1,4,6,5)(2,11,12,42)(3,16)(7,32,29,33)(8,25,9,28)(10,36)(13,46,44,59)(14,43,15,48)(17,20)(18,55)(19,23)(21,51)(22,47,26,30)(24,34,27,35)(31,53)(37,39)(38,52)(40,63)(41,62)(45,58)(49,54)(50,57)(56,61)(60,64);(1,5,6,4)(2,11,12,42)(3,17,21,23)(7,33,29,32)(8,28,9,25)(10,37,41,63)(13,46,44,59)(14,43,15,48)(16,20,51,19)(18,50,31,61)(22,47,26,30)(24,35,27,34)(36,39,62,40)(38,64,45,49)(52,60,58,54)(53,56,55,57)

Polycyclic presentation:

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

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, 416539);