#9039137

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

4147 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)(3,4,9,7)(5,13,14,10)(11,15,16,12)(17,18)(19,24)(20,28)(21,26)(22,23)(25,29)(27,32)(30,31)(33,34)(35,43)(36,40)(37,41)(38,42)(39,45)(44,48)(46,47);(1,3)(2,7)(4,8)(5,11)(6,9)(10,15)(12,13)(14,16)(17,19,23,21)(18,24,22,26)(20,27,29,30)(25,31,28,32)(33,35,38,45)(34,39,42,43)(36,41,46,44)(37,40,48,47);(2,8)(4,7)(5,14)(11,16)(18,22)(19,21)(25,28)(27,30)(36,46)(37,48)(40,47)(41,44);(1,3)(2,4)(5,11)(6,9)(7,8)(10,12)(13,15)(14,16)(17,20,23,29)(18,25,22,28)(19,27,21,30)(24,31,26,32)(33,34)(35,39)(36,47)(37,41)(38,42)(40,46)(43,45)(44,48);(1,4)(2,9)(3,8)(5,12)(6,7)(10,16)(11,13)(14,15)(17,21,23,19)(18,26,22,24)(20,27,29,30)(25,31,28,32)(33,36,38,46)(34,40,42,47)(35,44,45,41)(37,43,48,39);(1,5,6,14)(2,10,8,13)(3,11,9,16)(4,12,7,15)(17,22,23,18)(19,26,21,24)(20,28,29,25)(27,32,30,31)(33,37)(34,41)(35,40)(36,43)(38,48)(39,46)(42,44)(45,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^2=i,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,e^c=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, 8946333);