Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10012 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #127 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C23 | #392 |
| prime radical | C2×#3100 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1320 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(6,10)(7,8)(11,12)(13,14,18,21,19,23,17,22)(15,24,25,16,26,27,20,28);(2,4)(5,6)(7,11)(8,10)(9,12)(13,15)(14,16)(17,20)(18,25)(19,26)(21,27)(22,24)(23,28);(6,10)(7,8)(11,12)(13,16)(14,20)(15,21)(17,24)(18,27)(19,28)(22,26)(23,25)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=hi,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,c^b=cf,d^b=dh,f^b=fh,d^c=di,e^c=ehi,f^c=fh80 conjugacy classes — view the full character table.
| maximal subgroup of | #72211654, C2×#115424, #72237257, … |
| maximal normal subgroup of | #72211654, C2×#115424, #72237257, … |
| maximal quotient of | #72211633, C2×#115424, #72216508, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 22620);