#9053872

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

4477 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8961068);