Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432861 |
| 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 | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3004 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(7,11)(10,12)(14,20)(15,22)(17,18)(21,27)(23,28)(25,26);(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,19,22)(15,18,20,17)(16,21,24,28)(23,26,27,25);(6,9)(10,12)(13,15)(14,18)(16,23)(17,22)(19,20)(21,26)(24,27)(25,28);(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,23)(17,25)(18,26)(19,24)(20,27)(22,28);(16,24)(21,28)(23,27)(25,26)Polycyclic presentation:
2^9|a^2=fg,b^2=i,d^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,c^b=ch,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6342652);