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 | |
3190 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)(14,19)(15,22)(18,21)(20,26)(23,24)(25,28);(2,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,23)(16,18)(17,19)(20,25)(21,27)(22,26)(24,28);(5,7)(6,9)(8,12)(10,11)(13,15,17,25)(14,20,19,23)(16,26,27,24)(18,28,21,22);(7,12)(9,11)(13,16)(14,21)(15,24)(17,27)(18,19)(20,28)(22,23)(25,26);(6,10)(9,11)(14,19)(16,27)(20,23)(24,26)Polycyclic presentation:
2^9|a^2=fi,c^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=dh,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, 6407370);