Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #34743 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2182 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,11)(8,10)(9,12)(13,14,19,23,18,15,20,26)(16,24,17,25,27,21,28,22);(5,7)(6,9)(8,12)(10,11)(13,15,19,26,18,14,20,23)(16,24,17,25,27,21,28,22);(1,2,3,4)(13,16,18,27)(14,21,15,24)(17,20,28,19)(22,26,25,23);(7,12)(9,11)(13,17)(14,22)(15,25)(16,20)(18,28)(19,27)(21,26)(23,24);(6,10)(9,11)(16,27)(17,28)(21,24)(22,25)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,d^b=df,e^c=ei,e^d=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6296451);