Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| 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 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2547 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)(7,9)(8,10)(11,12)(14,20)(15,23)(16,26)(17,19)(21,27)(24,25);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,14)(15,21)(16,25)(17,28)(18,20)(19,22)(23,26)(24,27);(2,4)(5,8,9,12)(6,10,7,11)(13,15,18,25)(14,21,20,16)(17,23,19,24)(22,27,28,26);(6,7)(10,11)(13,16)(14,15)(17,27)(18,21)(19,26)(20,25)(22,24)(23,28);(13,17,18,19)(14,22,20,28)(15,24,25,23)(16,27,21,26)Polycyclic presentation:
2^9|b^2=fi,c^2=f,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,e^c=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, 7409005);