#147974

order	512
nilpotency class	3
isoclinism	isoclinic class of #2008
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#392
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11042
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

667 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,e^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fi,c^b=cf,f^b=fi,d^c=di,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55170);