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 |
| prime radical | #24372 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2165 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,11)(8,9)(10,12)(13,14,19,24)(15,25,27,21)(16,28,26,20)(17,22,18,23);(1,2)(3,4)(6,9)(10,11)(15,26)(16,27)(17,18)(20,25)(21,28)(22,23);(13,15)(14,20)(16,17)(18,26)(19,27)(21,23)(22,25)(24,28);(2,4)(13,16,19,26)(14,21,24,25)(15,17,27,18)(20,23,28,22);(5,7,8,12)(6,10,9,11)(13,17,19,18)(14,22,24,23)(15,16,27,26)(20,25,28,21)Polycyclic presentation:
2^9|a^2=i,d^2=i,e^2=fi,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 | #72308615, #72308763 |
| maximal normal subgroup of | #72308615, #72308763 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7425835);