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×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2139 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)(7,9)(8,10)(11,12)(13,14,19,24)(15,20,25,27)(16,21,18,23)(17,22,26,28);(1,2,4,3)(5,7)(6,9)(8,11)(10,12)(13,15)(14,20)(16,17)(18,26)(19,25)(21,22)(23,28)(24,27);(6,10)(7,11)(13,16,19,18)(14,21,24,23)(15,17,25,26)(20,22,27,28);(6,10)(9,12)(13,17,16,15,19,26,18,25)(14,22,21,20,24,28,23,27);(1,3,4,2)(13,16,19,18)(14,21,24,23)(15,17,25,26)(20,22,27,28)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,d^2=hi,e^2=fi,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.
| maximal subgroup of | #72308261, #72308843 |
| maximal normal subgroup of | #72308261, #72308843 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7519208);