Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884969 |
| 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 | #2615 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1481 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)(3,4)(5,6)(7,8)(17,18,21,26)(19,28,30,23)(20,32,34,25)(22,24,36,27)(29,38,33,41)(31,39,44,37)(35,42,46,40)(43,47,45,48);(2,6)(4,7)(9,10)(11,15)(12,14)(13,16)(17,19,21,30)(18,23,26,28)(20,33,34,29)(22,31,36,44)(24,37,27,39)(25,41,32,38)(35,45,46,43)(40,48,42,47);(1,3)(2,7)(4,6)(5,8)(9,11)(10,13)(12,16)(14,15)(18,24)(20,34)(22,36)(23,37)(25,42)(26,27)(28,39)(29,33)(31,44)(32,40)(38,48)(41,47);(2,6)(3,8)(11,16)(13,15)(17,20)(18,25)(19,29)(21,34)(22,35)(23,38)(24,40)(26,32)(27,42)(28,41)(30,33)(31,43)(36,46)(37,47)(39,48)(44,45);(1,4,5,7)(2,3,6,8)(10,14)(13,15)(20,34)(25,32)(29,33)(35,46)(38,41)(40,42)(43,45)(47,48)Polycyclic presentation:
2^9|a^2=i,b^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,d^c=dfh,g^c=gi,e^d=ei59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6796798);