Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514196 |
| 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 | #36152 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3342 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,11)(8,10)(9,12)(13,14,16,20)(15,22,24,18)(17,19,26,21)(23,28,25,27);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,15,16,24)(14,18,20,22)(17,23,26,25)(19,27,21,28);(14,19)(15,23)(17,26)(20,21)(24,25)(27,28);(2,4)(14,19)(15,23)(17,26)(20,21)(24,25)(27,28);(6,10)(9,11)(14,20)(15,24)(19,21)(23,25)Polycyclic presentation:
2^9|a^2=i,b^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,g^b=gi,h^b=hi,g^c=gi,h^c=hi,g^d=gi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72282583, #72282743, #72283017, … |
| maximal normal subgroup of | #72282583, #72282743, #72283017, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7421704);