Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2328 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)(8,12)(9,10)(13,14,19,23)(15,24,26,21)(16,27,25,28)(17,20,22,18);(1,2)(3,4)(6,10)(11,12)(13,15,19,26)(14,21,23,24)(16,18,25,20)(17,27,22,28);(5,7,9,8)(6,11,10,12)(13,16,20,15,19,25,18,26)(14,21,17,28,23,24,22,27);(2,4)(5,8,9,7)(6,12,10,11)(13,17,18,14,19,22,20,23)(15,24,16,28,26,21,25,27);(6,10)(11,12)(13,18,19,20)(14,22,23,17)(15,25,26,16)(21,28,24,27)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fg,d^2=fgi,e^2=i,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^c=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72308304, #72308510, #72308754, … |
| maximal normal subgroup of | #72308304, #72308510, #72308754, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7469374);