Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #35251 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2888 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,11)(8,10)(9,12)(13,14,17,23)(15,25,27,19)(16,24,18,22)(20,28,21,26);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15,17,27)(14,19,23,25)(16,28,18,26)(20,22,21,24);(2,4)(13,15)(14,20)(16,26)(17,27)(18,28)(19,24)(21,23)(22,25);(6,10)(7,12)(13,15)(14,21)(16,26)(17,27)(18,28)(19,22)(20,23)(24,25);(13,16,17,18)(14,22,23,24)(15,26,27,28)(19,20,25,21)Polycyclic presentation:
2^9|a^2=i,b^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,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, 7407884);