#149410

order	512
nilpotency class	4
isoclinism	isoclinic class of #140091
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1918
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11379
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

443 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=deh,c^2=deh,d^2=h,e^2=h,f^2=hi,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,d^b=dh,e^b=eh,f^b=fh,g^b=gh,h^b=hi,d^c=dh,e^c=eh,f^c=fh,g^c=ghi,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, 56606);