#8534683

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

4355 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=g,c^2=i,e^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8441879);