Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 |
| prime radical | #35728 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2376 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,11)(8,10)(9,12);(5,7)(6,9)(8,12)(10,11)(13,14,17,21)(15,19,22,27)(16,18,24,20)(23,25,28,26);(1,2,3,4)(14,18)(16,24)(19,25)(20,21)(23,28)(26,27);(6,10)(9,11)(13,15)(14,19)(16,23)(17,22)(18,25)(20,26)(21,27)(24,28);(7,12)(9,11)(14,18)(15,22)(16,24)(19,26)(20,21)(25,27)Polycyclic presentation:
2^9|b^2=i,c^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72282982, #72283891, #72284087, … |
| maximal normal subgroup of | #72282982, #72283891, #72284087, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6589398);