Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7200042 |
| 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 | #25578 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1568 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)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,24)(19,29,32,42)(20,33,35,26)(21,28,23,27)(25,37,38,31)(30,44,46,43)(34,40,36,41)(39,47,48,45);(1,3,6,2)(4,7,5,8)(9,11)(10,13)(12,15)(14,16)(18,24)(25,38)(26,33)(27,28)(29,42)(39,48)(40,41)(43,44);(11,15)(13,16)(17,19,22,32)(18,25,24,38)(20,30,35,46)(21,31,23,37)(26,39,33,48)(27,29,28,42)(34,45,36,47)(40,44,41,43);(1,4,6,5)(2,7,3,8)(10,14)(11,15)(17,20)(18,26)(19,30)(21,36)(22,35)(23,34)(24,33)(25,39)(27,41)(28,40)(29,43)(31,47)(32,46)(37,45)(38,48)(42,44);(1,5,6,4)(2,7,3,8)(11,15)(13,16)(17,21,22,23)(18,27,24,28)(19,31,32,37)(20,34,35,36)(25,29,38,42)(26,40,33,41)(30,45,46,47)(39,44,48,43)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=i,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dhi,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,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, 7113576);