Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| 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×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2039 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,10)(7,9,11,12)(13,14,18,23)(15,24,25,20)(16,26,27,21)(17,22,19,28);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15,18,25)(14,20,23,24)(16,17,27,19)(21,22,26,28);(2,4)(5,7,8,11)(6,9,10,12)(13,16,18,27)(14,20,23,24)(15,17,25,19)(21,28,26,22);(6,10)(9,12)(13,15)(14,21)(16,17)(18,25)(19,27)(20,28)(22,24)(23,26);(13,17,18,19)(14,22,23,28)(15,16,25,27)(20,21,24,26)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=df,e^b=ei,g^c=gi,g^d=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408708);