Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2142 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)(7,11)(10,12)(13,14,20,25)(15,21,27,16)(17,22,26,28)(18,24,19,23);(1,2)(3,4)(6,9)(7,11)(13,15)(14,21)(16,25)(17,18)(19,26)(20,27)(22,24)(23,28);(5,6)(7,10)(8,9)(11,12)(13,16,18,22,20,21,19,28)(14,17,23,27,25,26,24,15);(2,4)(6,9)(10,12)(13,17,20,26)(14,22,25,28)(15,18,27,19)(16,23,21,24);(5,7,8,11)(6,10,9,12)(13,18,20,19)(14,23,25,24)(15,26,27,17)(16,28,21,22)Polycyclic presentation:
2^9|a^2=i,c^2=gi,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7061953);