#8826243

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

4322 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:

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

Polycyclic presentation:

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

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, 8733439);