Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514000 |
| 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×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2032 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,11)(8,12)(9,10)(13,14)(15,24)(16,23)(17,19)(18,22)(20,25)(21,27)(26,28);(1,2)(3,4)(6,10)(11,12)(14,19)(15,25)(16,18)(20,28)(21,24)(26,27);(13,15)(14,20)(16,25)(17,26)(18,27)(19,21)(22,24)(23,28);(2,4)(5,7,9,8)(6,11,10,12)(13,15)(14,21)(16,25)(17,26)(18,27)(19,20)(22,28)(23,24);(5,8,9,7)(6,11,10,12)(13,16,17,18)(14,22,19,23)(15,25,26,27)(20,24,21,28)Polycyclic presentation:
2^9|d^2=fi,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,g^b=gi,h^b=hi,g^c=gi,h^c=hi,g^d=gi,h^d=hi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72307757, #72308081, #72308249 |
| maximal normal subgroup of | #72307757, #72308081, #72308249 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7428509);