Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667836 |
| 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 ◁ C4×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3086 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)(13,14)(15,23)(16,26)(17,19)(18,22)(20,24)(21,27)(25,28);(5,7)(6,9)(8,12)(10,11)(14,19)(15,16)(18,25)(20,26)(21,23)(24,27);(2,4)(13,15)(14,20)(16,25)(17,24)(18,27)(19,23)(21,28)(22,26);(6,10)(9,11)(14,19)(15,24)(16,27)(22,28);(7,12)(9,11)(13,16,17,27)(14,21,19,26)(15,25,24,18)(20,28,23,22)Polycyclic presentation:
2^9|e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72281997, #72282112, #72282448, … |
| maximal normal subgroup of | #72281997, #72282112, #72282448, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6575065);