Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 | #44899 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2083 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,10)(7,9,11,12)(13,14,19,24)(15,16,21,22)(17,20,27,25)(18,26,28,23);(1,2)(3,4)(5,7)(6,9)(8,11)(10,12)(13,15,18,17,19,21,28,27)(14,20,23,16,24,25,26,22);(6,10)(7,11)(13,16)(14,21)(15,24)(17,26)(18,25)(19,22)(20,28)(23,27);(2,4)(6,10)(9,12)(15,21)(16,22)(17,27)(20,25);(13,17,28,15,19,27,18,21)(14,22,26,25,24,16,23,20)Polycyclic presentation:
2^9|a^2=gi,b^2=f,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72281938, #72282009 |
| maximal normal subgroup of | #72281938, #72282009 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7434780);