Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574015 |
| 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 | #39436 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2098 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,12)(8,10)(9,11)(13,14,18,22)(15,19,20,28)(16,24,27,23)(17,25,26,21);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15)(14,19)(16,25)(17,23)(18,20)(21,27)(22,28)(24,26);(2,4)(13,16,26,28,18,27,17,19)(14,20,25,23,22,15,21,24);(1,2)(3,4)(5,8,9,7)(6,10,11,12)(15,20)(16,27)(19,28)(23,24);(2,4)(15,20)(16,27)(19,28)(23,24)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gi,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,e^c=ei,e^d=ehi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6491459);