Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514000 |
| 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 |
| prime radical | C2×#127 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1789 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6,9,7)(8,11,12,10)(13,14,18,22)(15,19,17,21)(16,20,24,27)(23,26,25,28);(1,2,4,3)(5,7,9,6)(8,10,12,11)(15,17)(16,23)(19,21)(20,26)(24,25)(27,28);(5,8,9,12)(6,10,7,11)(13,15,18,17)(14,19,22,21)(16,23,24,25)(20,26,27,28);(6,7)(10,11)(13,16,18,24)(14,20,22,27)(15,23,17,25)(19,26,21,28);(1,3,4,2)(13,15,18,17)(14,19,22,21)(16,23,24,25)(20,26,27,28)Polycyclic presentation:
2^9|a^2=gi,b^2=fg,c^2=gi,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,e^b=ei,h^b=hi,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7452954);