Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518944 |
| 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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2163 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)(7,10)(8,9)(11,12)(13,14,18,23)(15,24,19,20)(16,17,26,27)(21,25,28,22);(2,4)(6,9)(10,12)(13,15)(14,20)(16,26)(18,19)(21,28)(23,24);(5,7,8,11)(6,10,9,12)(13,16,15,25,18,26,19,22)(14,21,20,27,23,28,24,17);(1,2,3,4)(13,17,19,28,18,27,15,21)(14,22,24,26,23,25,20,16);(6,9)(10,12)(14,23)(17,27)(20,24)(21,28)Polycyclic presentation:
2^9|a^2=i,c^2=fg,d^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72257078, #72257119, #72257163 |
| maximal normal subgroup of | #72257078, #72257119, #72257163 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7520786);