Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573485 |
| 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 | C2×#3268 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2267 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)(7,10)(8,9)(11,12)(13,14,18,22)(15,19,16,20)(17,21,25,28)(23,26,24,27);(6,9)(10,12)(13,15)(14,19)(16,18)(17,23)(20,22)(21,26)(24,25)(27,28);(2,4)(5,7,8,11)(6,10,9,12)(15,23)(16,24)(17,25)(19,26)(20,27)(21,28);(13,16,18,15)(14,20,22,19)(17,24,25,23)(21,27,28,26);(15,23)(16,24)(17,25)(19,26)(20,27)(21,28)Polycyclic presentation:
2^9|a^2=i,c^2=f,d^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72308805, #72308822 |
| maximal normal subgroup of | #72308805, #72308822 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6482032);