Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| 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 | C2×#3496 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4373 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)(7,9,11,12)(13,14,19,16)(15,22,25,28)(17,21,18,26)(20,27,23,24);(5,7)(6,9)(8,11)(10,12)(13,15)(14,20)(16,23)(17,24)(18,27)(19,25)(21,28)(22,26);(2,4)(7,11)(9,12)(13,16,19,14)(15,23,25,20)(17,21,18,26)(22,24,28,27);(5,7)(6,9)(8,11)(10,12)(14,16)(20,23)(21,26)(22,28);(7,11)(9,12)(13,17,19,18)(14,21,16,26)(15,24,25,27)(20,28,23,22)Polycyclic presentation:
2^9|a^2=hi,c^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,e^d=eh77 conjugacy classes — view the full character table.
| maximal subgroup of | #72308707, #72308792 |
| maximal normal subgroup of | #72308707, #72308792 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6498654);