#8943756

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

4100 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)(3,7,11,15)(4,9,12,14)(5,10,13,16)(17,18,22,25)(19,23,29,32)(20,24,30,21)(26,27,28,31)(33,34,39,37)(35,36,43,40)(38,45,47,41)(42,48,46,44);(1,3)(2,7)(4,5)(6,11)(8,15)(9,10)(12,13)(14,16)(17,19)(18,23)(20,28)(21,27)(22,29)(24,31)(25,32)(26,30)(33,35,39,43)(34,40,37,36)(38,42,47,46)(41,48,45,44);(2,8)(3,11)(5,13)(9,14)(17,20,22,30)(18,24,25,21)(19,26,29,28)(23,27,32,31)(33,35,39,43)(34,40,37,36)(38,46,47,42)(41,44,45,48);(1,4)(2,9)(3,5)(6,12)(7,10)(8,14)(11,13)(15,16)(17,21)(18,20)(19,27)(22,24)(23,28)(25,30)(26,32)(29,31)(33,36)(34,35)(37,43)(38,44)(39,40)(41,42)(45,46)(47,48);(1,5)(2,10)(3,12)(4,11)(6,13)(7,14)(8,16)(9,15)(17,20)(18,24)(19,28)(21,25)(22,30)(23,31)(26,29)(27,32)(33,37,39,34)(35,40,43,36)(38,45,47,41)(42,48,46,44);(1,2)(3,7)(4,9)(5,10)(6,8)(11,15)(12,14)(13,16)(17,19)(18,23)(20,28)(21,27)(22,29)(24,31)(25,32)(26,30)(33,38,39,47)(34,41,37,45)(35,42,43,46)(36,44,40,48)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8850952);