Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438991 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2497 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)(7,10)(8,9)(11,12)(13,14,20,17,19,15,18,22)(16,21,28,24,26,23,27,25);(2,4)(13,15,19,14)(16,24,26,25)(17,18,22,20)(21,27,23,28);(6,9)(10,12)(13,16,18,27,19,26,20,28)(14,21,22,25,15,23,17,24);(13,17,18,14,19,22,20,15)(16,25,27,23,26,24,28,21);(5,7,8,11)(6,10,9,12)(16,26)(21,23)(24,25)(27,28)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=h,d^2=h,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,d^c=di,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72307067, #72307108, #72307186, … |
| maximal normal subgroup of | #72307067, #72307108, #72307186, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6374735);