Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518952 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 ◁ C4×#127 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2047 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6)(7,11)(8,9)(10,12)(13,14,18,21)(15,24,25,20)(16,27,26,22)(17,28,19,23);(2,3)(6,9)(10,11)(13,15,18,25)(14,20,21,24)(16,17,26,19)(22,28,27,23);(1,2,4,3)(14,21)(15,25)(16,26)(23,28);(1,3,4,2)(13,16,19,15,18,26,17,25)(14,22,23,20,21,27,28,24);(5,7,8,12)(6,10,9,11)(13,17,18,19)(14,23,21,28)(15,26,25,16)(20,22,24,27)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=gi,d^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7521014);