Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6629007 |
| 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 | #2847 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2461 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,17,20)(15,21,22,28)(16,24,25,19)(18,26,27,23);(6,9)(10,12);(13,15,16,23,17,22,25,26)(14,18,19,21,20,27,24,28);(2,4);(5,7,8,11)(6,10,9,12)(15,22)(18,27)(21,28)(23,26)Polycyclic presentation:
2^9|a^2=i,c^2=g,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72238725, #72249378, #72249470, … |
| maximal normal subgroup of | #72238725, #72249378, #72249470, … |
| maximal quotient of | #72239782, #72264864, #72307790, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6537504);