Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437529 |
| 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 | #23390 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2228 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14,18,21)(15,19,24,27)(16,22,17,26)(20,25,28,23);(1,2)(3,4)(6,9)(10,12)(13,15)(14,19)(16,25)(17,23)(18,24)(20,26)(21,27)(22,28);(2,4)(13,15)(14,20)(16,23)(17,25)(18,24)(19,26)(21,28)(22,27);(2,4)(5,7,8,11)(6,10,9,12)(14,21)(19,27)(20,28)(22,26);(13,16,18,17)(14,22,21,26)(15,23,24,25)(19,20,27,28)Polycyclic presentation:
2^9|a^2=i,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,d^b=dh,e^b=ei,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72307385, #72307495, #72307670 |
| maximal normal subgroup of | #72307385, #72307495, #72307670 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346513);