Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395154 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1610 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6,7,8)(13,14)(15,25)(16,28)(17,24)(18,21)(19,23)(20,27)(22,26);(1,2,3,4)(9,10)(11,12)(13,15,17,27,18,26,19,16)(14,20,23,25,21,28,24,22);(6,8)(10,12)(14,21)(15,26)(16,27)(23,24);(13,16,19,26,18,27,17,15)(14,22,24,28,21,25,23,20)Polycyclic presentation:
2^9|a^2=fi,b^2=e,d^2=h,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72264550, #72264562, #72264595, … |
| maximal normal subgroup of | #72264550, #72264562, #72264595, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 319602);