Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #158815 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C25 | C24 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C25 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
4040 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,7,8)(10,12)(13,14,17,15)(16,18,20,19)(23,27)(25,28);(2,4)(10,12)(14,15)(16,20)(21,22,24,26)(23,28,27,25);(9,10)(11,12)(13,15,17,14)(16,18,20,19)(21,22)(23,25)(24,26)(27,28);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,16)(14,18)(15,19)(17,20)(21,23)(22,25)(24,27)(26,28)Polycyclic presentation:
2^9|a^2=gh,b^2=efg,c^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,d^c=de92 conjugacy classes — view the full character table.
| maximal subgroup of | #72328396, #72328453, #72328663, … |
| maximal normal subgroup of | #72328396, #72328453, #72328663, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 66449);