Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7200022 |
| 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 | #36085 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2226 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,27)(19,28,29,40)(20,32,33,25)(21,26,22,36)(24,37,38,31)(30,45,44,43)(34,42,35,41)(39,47,48,46);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(19,29)(20,33)(24,38)(25,32)(28,40)(31,37)(34,35)(41,42);(11,15)(13,16)(17,19,23,29)(18,24,27,38)(20,30,33,44)(21,31,22,37)(25,39,32,48)(26,40,36,28)(34,46,35,47)(41,43,42,45);(3,8)(5,7)(10,14)(11,15)(17,20)(18,25)(19,30)(21,35)(22,34)(23,33)(24,39)(26,42)(27,32)(28,43)(29,44)(31,47)(36,41)(37,46)(38,48)(40,45);(2,6)(5,7)(11,15)(13,16)(17,21,23,22)(18,26,27,36)(19,31,29,37)(20,34,33,35)(24,40,38,28)(25,41,32,42)(30,46,44,47)(39,43,48,45)Polycyclic presentation:
2^9|a^2=i,c^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,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, 7107391);