#9066104

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

4197 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)(4,5)(6,14)(7,8)(10,15)(11,13)(12,16)(18,23)(19,25)(24,31)(26,28)(33,34,39,43)(35,40,44,46)(36,38,42,48)(37,41,47,45);(1,3,7,4)(2,5,8,9)(6,11,15,12)(10,13,14,16)(17,18,22,23)(19,21,25,30)(20,24,27,31)(26,29,28,32)(33,35)(34,40)(36,45)(37,38)(39,44)(41,42)(43,46)(47,48);(2,8)(5,9)(10,14)(13,16)(20,27)(24,31)(26,28)(29,32)(35,44)(37,47)(40,46)(41,45);(1,4,7,3)(2,9,8,5)(6,11,15,12)(10,13,14,16)(17,19)(18,21)(20,28)(22,25)(23,30)(24,32)(26,27)(29,31)(33,36)(34,38)(35,45)(37,40)(39,42)(41,44)(43,48)(46,47);(1,5,7,9)(2,4,8,3)(6,13,15,16)(10,12,14,11)(17,20,22,27)(18,24,23,31)(19,26,25,28)(21,29,30,32)(33,37)(34,41)(35,38)(36,46)(39,47)(40,42)(43,45)(44,48);(1,6)(2,10)(3,11)(4,12)(5,13)(7,15)(8,14)(9,16)(17,21)(18,25)(19,23)(20,29)(22,30)(24,28)(26,31)(27,32)(33,38)(34,42)(35,37)(36,43)(39,48)(40,41)(44,47)(45,46)

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8973300);