Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18192 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1678 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6,8,10,9,12,7,11)(13,14,18,22,19,15,17,21)(16,24,25,23,27,28,26,20);(1,2,3,4)(6,10)(7,8)(11,12)(13,15)(14,19)(16,23)(17,21)(18,22)(20,27)(24,25)(26,28);(6,10)(7,8)(11,12)(13,16)(14,20)(15,23)(17,26)(18,25)(19,27)(21,28)(22,24);(5,7,9,8)(6,11,12,10)(13,17,19,18)(14,21,15,22)(16,25,27,26)(20,24,23,28)Polycyclic presentation:
2^9|a^2=g,b^2=eghi,d^2=h,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,f^a=fhi,d^b=dh,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,f^c=fh,g^c=gi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72250407, #72250419, #72250438, … |
| maximal normal subgroup of | #72250407, #72250419, #72250438, … |
| maximal quotient of | #72250929, #72250955, #72250959, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 102403);