Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | #2011 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C4×C4⋊C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
777 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,4,3)(5,6,9,12)(7,10,8,11)(13,14,17,21,18,16,25,20)(15,19,23,27,24,22,28,26);(2,3)(5,7,9,8)(6,10,12,11)(13,15,17,23,18,24,25,28)(14,19,21,27,16,22,20,26);(6,11)(7,8)(10,12)(13,16)(14,18)(15,22)(17,20)(19,24)(21,25)(23,26)(27,28);(1,3,4,2)(14,16)(19,22)(20,21)(26,27)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,c^b=chi,d^b=de,d^c=di,f^c=fh92 conjugacy classes — view the full character table.
| maximal subgroup of | #72343120, #72343139, #72343143, … |
| maximal normal subgroup of | #72343120, #72343139, #72343143, … |
| maximal quotient of | #72343441, #72343445, #72343469, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 136986);