Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | #45245 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1664 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,26)(19,27,28,39)(20,24,35,42)(21,36,29,25)(23,31,37,32)(30,43,44,48)(33,41,34,40)(38,46,47,45);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(19,28)(23,37)(27,39)(30,44)(31,32)(38,47)(43,48)(45,46);(1,3,4,7)(2,6,5,8)(9,11)(10,13)(12,16)(14,15)(17,19,29,31,22,28,21,32)(18,23,36,39,26,37,25,27)(20,30,33,46,35,44,34,45)(24,38,40,48,42,47,41,43);(2,5)(6,8)(11,16)(13,15)(17,20)(18,24)(19,30)(21,34)(22,35)(23,38)(25,41)(26,42)(27,43)(28,44)(29,33)(31,46)(32,45)(36,40)(37,47)(39,48);(10,14)(13,15)(17,21,22,29)(18,25,26,36)(19,31,28,32)(20,33,35,34)(23,39,37,27)(24,40,42,41)(30,45,44,46)(38,43,47,48)Polycyclic presentation:
2^9|a^2=i,c^2=fgi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6927000);