Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #45136 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1593 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,18,23,36,22,28,21,27)(19,24,37,47,32,41,31,40)(20,26,33,25,35,42,34,43)(29,44,45,38,30,39,46,48);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,29)(21,31)(22,32)(23,37)(25,38)(26,44)(27,40)(28,41)(30,35)(33,45)(34,46)(36,47)(39,42)(43,48);(1,2,5,4)(3,6,8,7)(9,11)(10,13)(12,16)(14,15)(17,20)(18,25)(19,29)(21,33)(22,35)(23,34)(24,38)(26,36)(27,42)(28,43)(30,32)(31,45)(37,46)(39,40)(41,48)(44,47);(1,2,5,4)(3,7,8,6)(11,16)(13,15)(17,20)(18,26)(19,30)(21,34)(22,35)(23,33)(24,39)(25,36)(27,43)(28,42)(29,32)(31,45)(37,46)(38,40)(41,44)(47,48);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(17,21,22,23)(18,27,28,36)(19,31,32,37)(20,33,35,34)(24,40,41,47)(25,42,43,26)(29,45,30,46)(38,39,48,44)Polycyclic presentation:
2^9|a^2=fgi,c^2=f,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6924356);