Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6629009 |
| 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 |
| prime radical | #2615 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2181 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)(5,6)(7,10)(8,9)(11,12)(13,14,18,22)(15,23,25,20)(16,17,21,19)(24,27,28,26);(6,9)(10,12)(13,15)(14,20)(16,24)(17,26)(18,25)(19,27)(21,28)(22,23);(13,16,19,14,18,21,17,22)(15,24,27,20,25,28,26,23);(2,4)(15,25)(20,23)(24,28)(26,27);(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,21,22,16)(15,26,25,27)(20,28,23,24)Polycyclic presentation:
2^9|a^2=i,c^2=g,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,d^b=di,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72263954, #72263990, #72264011 |
| maximal normal subgroup of | #72263954, #72263990, #72264011 |
| maximal quotient of | #72264864 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6537510);