Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499792 |
| 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×#3207 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2138 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(13,14,18,23)(15,19,24,27)(16,22,17,21)(20,26,28,25);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15,18,24)(14,19,23,27)(16,26,17,25)(20,22,28,21);(6,9)(7,11)(13,15)(14,20)(16,25)(17,26)(18,24)(19,21)(22,27)(23,28);(5,7,8,11)(6,10,9,12)(15,24)(19,27)(20,28)(25,26);(6,9)(10,12)(13,16,18,17)(14,21,23,22)(15,25,24,26)(19,28,27,20)Polycyclic presentation:
2^9|a^2=i,b^2=fi,d^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6420669);