#9029849

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

4044 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=i,d^2=gh,e^2=h,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8937045);