#8418600

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

4005 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gh,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=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, 8325796);