Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499823 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3347 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(13,14)(15,19)(16,21)(17,26)(18,22)(20,23)(24,28)(25,27);(2,4)(5,6)(7,11)(8,10)(9,12)(13,15,18,24)(14,19,22,28)(16,25,17,23)(20,21,27,26);(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,23)(17,25)(18,24)(19,26)(21,28)(22,27);(7,12)(9,11)(13,15)(14,19)(16,25)(17,23)(18,24)(20,26)(21,27)(22,28);(6,10)(9,11)(13,16,18,17)(14,21,22,26)(15,23,24,25)(19,20,28,27)Polycyclic presentation:
2^9|a^2=f,b^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6407186);