Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2458 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)(5,6)(7,9)(8,10)(11,12)(13,14,17,21)(15,19,22,27)(16,18,24,20)(23,25,28,26);(2,3)(5,6)(7,11)(8,10)(9,12);(5,7)(6,9)(8,12)(10,11)(14,18)(15,22)(16,24)(19,26)(20,21)(25,27);(1,3,4,2)(7,12)(9,11)(13,15)(14,19)(16,23)(17,22)(18,25)(20,26)(21,27)(24,28);(6,10)(9,11)(15,22)(19,27)(23,28)(25,26)Polycyclic presentation:
2^9|a^2=fi,d^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
| maximal subgroup of | #72281760, #72282387, #72282635, … |
| maximal normal subgroup of | #72281760, #72282387, #72282635, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6903723);