Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| 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 | #35680 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2969 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)(3,4)(5,6)(7,11)(8,10)(9,12);(5,7)(6,9)(8,12)(10,11)(13,14,17,21)(15,22,23,28)(16,19,26,20)(18,25,27,24);(2,4)(13,15)(14,18)(16,25)(17,23)(19,22)(20,28)(21,27)(24,26);(6,10)(9,11)(15,23)(18,27)(22,28)(24,25);(7,12)(9,11)(14,19)(15,23)(16,26)(18,28)(20,21)(22,27)Polycyclic presentation:
2^9|b^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72282961, #72283749, #72283996, … |
| maximal normal subgroup of | #72282961, #72283749, #72283996, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6574782);