Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7049 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,18,21)(15,19,24,28)(16,20,26,17)(22,27,25,23);(6,9)(10,12)(13,15,18,24)(14,19,21,28)(16,25,26,22)(17,27,20,23);(1,2,3,4)(13,16)(14,20)(15,22)(17,21)(18,26)(19,27)(23,28)(24,25);(2,4)(13,17,18,20)(14,16,21,26)(15,23,24,27)(19,22,28,25);(14,21)(16,26)(19,28)(22,25);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=h,d^2=i,f^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,d^c=dh,e^c=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7576472);