Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7339802 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #48088 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,12,15,26)(4,8)(5,10)(6,11)(7,14,9,19)(13,28,30,22)(16,21,31,24)(17,23)(18,25)(20,27)(29,32);(1,3)(2,7)(4,16)(5,19)(6,15)(8,22)(9,11)(10,26)(12,27)(13,29)(14,20)(17,31)(18,30)(21,32)(23,28)(24,25);(4,17)(5,20)(8,23)(10,27)(12,26)(13,30)(14,19)(21,24);(1,4,5,18,6,17,20,29)(2,8,10,25,11,23,27,32)(3,13,19,16,15,30,14,31)(7,21,26,22,9,24,12,28);(1,3)(2,9)(4,13)(5,19)(6,15)(7,11)(8,24)(10,12)(14,20)(16,18)(17,30)(21,23)(22,32)(25,28)(26,27)(29,31)Polycyclic presentation:
2^9|a^2=g,d^2=fg,f^2=i,b^a=bf,c^a=cg,e^a=efi,c^b=ch,d^b=dfh,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,h^d=hi,f^e=fi,g^e=gi53 conjugacy classes — view the full character table.
| maximal quotient of | #72318947, #72320018 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7246998);