Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500689 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2846 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)(13,14,20,26,19,16,18,22)(15,17,23,27,24,21,25,28);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,21)(16,17)(18,25)(19,24)(20,23)(22,27)(26,28);(2,4)(13,16,19,14)(15,17,24,21)(18,22,20,26)(23,28,25,27);(6,10)(7,12)(13,17)(14,15)(16,24)(18,27)(19,21)(20,28)(22,23)(25,26);(13,18,19,20)(14,22,16,26)(15,23,24,25)(17,27,21,28)Polycyclic presentation:
2^9|a^2=gi,c^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,e^b=ei,e^c=ei,g^c=gi,g^d=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410428);