Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 | #35691 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2465 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)(7,11)(8,10)(9,12)(13,14,18,22)(15,16,24,25)(17,20,28,21)(19,27,23,26);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15)(14,16)(17,23)(18,24)(19,28)(20,26)(21,27)(22,25);(13,16)(14,19)(15,21)(17,27)(18,25)(20,24)(22,23)(26,28);(2,4)(5,7,9,8)(6,11,12,10)(14,20)(15,23)(17,28)(19,24)(21,22)(26,27);(5,8,9,7)(6,11,12,10)(15,24)(16,25)(19,23)(26,27)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=f,e^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=cgi,d^b=dh,e^b=ei,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72282919, #72283072, #72283488, … |
| maximal normal subgroup of | #72282919, #72283072, #72283488, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7429883);