Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7300980 |
| 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 | #2388 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1430 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,6)(3,7,8,5)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26)(27,35)(28,32)(29,41)(30,33)(31,37)(34,43)(36,49)(38,48)(39,46)(40,53)(42,45)(44,51)(47,52)(50,56)(54,55);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(28,39)(32,46)(36,50)(40,52)(44,55)(47,53)(49,56)(51,54);(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,27,30,38)(26,31,33,45)(28,36,39,50)(29,42,43,37)(32,44,46,55)(34,48,41,35)(40,54,52,51)(47,56,53,49);(3,8)(5,7)(11,18)(12,20)(14,22)(15,24)(25,28,30,39)(26,32,33,46)(27,36,38,50)(29,40,43,52)(31,44,45,55)(34,47,41,53)(35,49,48,56)(37,51,42,54);(2,6)(5,7)(11,18)(14,22)(17,19)(21,23)(26,33)(28,39)(31,45)(34,41)(35,48)(36,50)(40,52)(51,54)Polycyclic presentation:
2^9|a^2=fh,c^2=i,d^2=i,g^2=i,b^a=bf,c^a=cg,e^a=efi,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,d^c=dh,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, 7247266);