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)(15,21)(16,22)(17,23)(18,27)(19,25)(20,24)(26,28);(2,4)(5,6)(7,11)(8,10)(9,12)(13,15,17,25)(14,19,23,21)(16,26,18,24)(20,27,28,22);(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,24)(17,25)(18,26)(19,22)(21,27)(23,28);(7,12)(9,11)(13,15)(14,21)(16,26)(17,25)(18,24)(19,23)(20,27)(22,28);(6,10)(9,11)(13,16,17,18)(14,22,23,27)(15,24,25,26)(19,28,21,20)Polycyclic presentation:
2^9|a^2=fi,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, 6407187);