Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628851 |
| 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 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4306 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)(5,6,8,10)(7,11,12,9)(13,14,17,19)(15,21,24,28)(16,22,18,26)(20,25,27,23);(5,7,8,12)(6,9,10,11)(14,19)(20,27)(21,28)(22,26);(13,15)(14,20)(16,23)(17,24)(18,25)(19,27)(21,26)(22,28);(2,4)(13,15)(14,21)(16,25)(17,24)(18,23)(19,28)(20,26)(22,27);(13,16,17,18)(14,22,19,26)(15,23,24,25)(20,28,27,21)Polycyclic presentation:
2^9|a^2=f,b^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,g^c=gi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536290);