Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437521 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3499 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4126 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,11)(8,9)(10,12)(13,14,20,24)(15,18,23,19)(16,21,17,22)(25,27,26,28);(1,2,3,4)(6,9)(10,11)(13,15,18,24,20,23,19,14)(16,22,28,25,17,21,27,26);(5,7,8,12)(6,10,9,11)(13,16)(14,21)(15,25)(17,20)(18,27)(19,28)(22,24)(23,26);(13,17)(14,22)(15,26)(16,20)(18,28)(19,27)(21,24)(23,25);(13,18,20,19)(14,15,24,23)(16,27,17,28)(21,25,22,26)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=g,e^2=i,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=dh,f^c=fi,h^c=hi,f^d=fi,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6353871);