Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61629 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2066 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,11)(8,10)(9,12)(14,20)(15,23)(17,19)(21,28)(24,27)(25,26);(1,2,4,3)(5,7)(6,9)(8,12)(10,11)(13,14,18,20)(15,21,25,27)(16,19,22,17)(23,24,26,28);(1,3,4,2)(13,15)(14,21)(16,24)(17,23)(18,25)(19,26)(20,27)(22,28);(7,12)(9,11)(13,16,18,22)(14,17,20,19)(15,24,25,28)(21,23,27,26);(6,10)(9,11)(13,17,18,19)(14,22,20,16)(15,23,25,26)(21,28,27,24)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=h,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^b=ei,g^c=gi,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6301855);