Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7309060 |
| 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 | #2011 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2402 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1305 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23)(25,26,31,33,30,36,29,35)(27,32,43,48,37,47,39,45)(28,34,44,49,41,52,42,51)(38,46,54,56,40,50,53,55);(1,2,4,5)(3,7,8,6)(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(25,27)(26,32)(28,40)(29,39)(30,37)(31,43)(33,48)(34,50)(35,45)(36,47)(38,41)(42,54)(44,53)(46,52)(49,55)(51,56);(1,2)(3,6)(4,5)(7,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(26,33)(27,37)(28,41)(29,31)(32,45)(34,51)(35,36)(46,56)(47,48)(49,52)(50,55)(53,54);(2,5)(6,7)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,28)(26,34)(27,38)(29,42)(30,41)(31,44)(32,46)(33,49)(35,51)(36,52)(37,40)(39,53)(43,54)(45,55)(47,50)(48,56);(1,3,4,8)(2,6,5,7)(10,18)(15,19)(16,17)(23,24)(27,37)(28,41)(32,47)(34,52)(39,43)(42,44)(45,48)(49,51)Polycyclic presentation:
2^9|a^2=gi,b^2=fhi,c^2=f,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,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, 7392325);