#9508843

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

4315 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9416039);