Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60753 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
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×#3038 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2532 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(6,9)(7,11)(13,14)(15,18)(16,24)(17,22)(19,21)(20,25)(23,27)(26,28);(1,3,4,2)(5,6,8,9)(7,10,11,12)(13,15,18,14)(16,20,26,23)(17,22,19,21)(24,27,28,25);(6,9)(10,12)(13,16)(14,20)(15,23)(17,25)(18,26)(19,27)(21,24)(22,28);(5,7,8,11)(6,10,9,12)(14,15)(20,23)(21,22)(24,28);(13,17,18,19)(14,21,15,22)(16,25,26,27)(20,24,23,28)Polycyclic presentation:
2^9|a^2=h,b^2=fh,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=cf,d^b=di,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72240346, #72261189, #72283332, … |
| maximal normal subgroup of | #72240346, #72261189, #72283332, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6283054);