Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2162 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,3)(5,6)(7,8)(9,10,11,12)(13,14,16,20)(15,22,24,18)(17,19,26,21)(23,28,25,27);(1,2,4,3)(13,15)(14,18)(16,24)(17,23)(19,27)(20,22)(21,28)(25,26);(6,8)(14,19)(15,23)(17,26)(20,21)(24,25)(27,28);(1,3,4,2)(14,20)(18,22)(19,21)(27,28)Polycyclic presentation:
2^9|a^2=h,b^2=ei,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,f^c=fi,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72240762, #72240764, #72240765, … |
| maximal normal subgroup of | #72240762, #72240764, #72240765, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 267338);