Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #195954 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C26 | C2×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2792 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6)(7,9,11,12)(8,10)(13,14,17,20,18,21,16,24)(15,19,22,27,23,28,25,26);(5,7,8,11)(6,9)(10,12)(13,15,16,22,18,23,17,25)(14,19,20,26,21,28,24,27);(2,4);(13,16,18,17)(14,20,21,24)(15,22,23,25)(19,26,28,27)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,e^b=eh,g^b=gh104 conjugacy classes — view the full character table.
| maximal subgroup of | #72309279, #72309281, #72309285, … |
| maximal normal subgroup of | #72309279, #72309281, #72309285, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 103155);