Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499823 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
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 | |
4034 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(13,14,19,16)(15,22,26,28)(17,21,18,27)(20,24,23,25);(5,6)(7,11)(8,10)(9,12)(13,15)(14,20)(16,23)(17,25)(18,24)(19,26)(21,28)(22,27);(2,4)(5,7)(6,9)(8,12)(10,11)(13,14)(15,23)(16,19)(17,27)(18,21)(20,26)(22,24)(25,28);(7,12)(9,11)(13,16,19,14)(15,23,26,20)(17,21,18,27)(22,25,28,24);(6,10)(9,11)(13,17,19,18)(14,21,16,27)(15,24,26,25)(20,22,23,28)Polycyclic presentation:
2^9|a^2=i,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6407086);