Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438955 |
| 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 | #35252 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2440 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)(7,9)(8,10)(11,12)(13,14,18,19,17,23,16,22)(15,21,24,28,25,20,26,27);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(14,19)(15,24)(16,18)(20,21)(22,23)(25,26);(2,4)(6,10)(9,12)(13,15,16,26,17,25,18,24)(14,20,22,28,23,21,19,27);(2,4)(13,15,16,26,17,25,18,24)(14,21,22,27,23,20,19,28);(15,25)(20,21)(24,26)(27,28)Polycyclic presentation:
2^9|a^2=fi,b^2=gi,c^2=f,d^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,e^d=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72280185, #72280468, #72280641, … |
| maximal normal subgroup of | #72280185, #72280468, #72280641, … |
| maximal quotient of | #72313163 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6348996);