Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518944 |
| 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 | C2×#496 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1829 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6,9,8)(7,10,12,11)(13,14,19,22)(15,18,21,17)(16,23,26,27)(20,24,28,25);(1,2,4,3)(5,7)(6,10)(8,11)(9,12)(13,15)(14,18)(16,20)(17,22)(19,21)(23,24)(25,27)(26,28);(5,8,9,6)(7,11,12,10)(13,16,17,24,19,26,18,25)(14,20,21,27,22,28,15,23);(7,12)(10,11)(13,15,18,22,19,21,17,14)(16,23,25,28,26,27,24,20);(1,3,4,2)(13,17,19,18)(14,15,22,21)(16,24,26,25)(20,23,28,27)Polycyclic presentation:
2^9|a^2=ghi,b^2=f,c^2=h,d^2=g,e^2=fi,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,g^b=gi,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72314172 |
| maximal normal subgroup of | #72314172 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7511448);