Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437529 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3128 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,8,10)(7,11,12,9)(13,14)(15,23)(16,22)(17,21)(18,20)(19,24)(25,28)(26,27);(1,2,3,4)(13,15,16,24,18,26,17,25)(14,19,21,23,20,28,22,27);(5,7,8,12)(6,9,10,11)(15,24)(16,17)(19,23)(21,22)(25,26)(27,28);(15,25)(16,17)(19,27)(21,22)(23,28)(24,26);(14,20)(19,28)(21,22)(23,27)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,h^a=hi,c^b=ch,d^b=dhi,f^c=fi,h^c=hi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72284252, #72305349, #72305367 |
| maximal normal subgroup of | #72284252, #72305349, #72305367 |
| maximal quotient of | #72305620 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6352940);