#10047494

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4095 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2)(3,7)(4,5)(6,8)(9,10)(11,16)(12,15)(13,14)(17,18)(19,21)(20,26)(22,24)(23,25)(27,30)(28,31)(29,32)(33,34)(35,39)(36,42)(37,40)(38,41)(43,47)(44,48)(45,46);(1,3,6,11)(2,7,8,16)(4,14,10,12)(5,13,9,15)(17,19,23,22)(18,24,25,21)(20,28,29,27)(26,30,32,31)(35,36)(39,42)(43,44)(47,48);(2,8)(3,11)(4,10)(13,15)(18,25)(20,29)(21,24)(27,28)(33,35,38,36)(34,39,41,42)(37,43,46,44)(40,47,45,48);(1,4,6,10)(2,9,8,5)(3,12,11,14)(7,13,16,15)(17,20,23,29)(18,26,25,32)(19,27,22,28)(21,30,24,31)(33,36,38,35)(34,39,41,42)(37,44,46,43)(40,47,45,48);(1,3)(2,7)(4,12)(5,15)(6,11)(8,16)(9,13)(10,14)(17,21,23,24)(18,19,25,22)(20,30,29,31)(26,27,32,28)(33,34)(35,42)(36,39)(37,45)(38,41)(40,46)(43,47)(44,48);(1,5,6,9)(2,10,8,4)(3,13,11,15)(7,12,16,14)(17,22,23,19)(18,21,25,24)(20,28,29,27)(26,30,32,31)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48)

Polycyclic presentation:

2^9|b^2=h,c^2=ghi,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9954690);