Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417575 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2002 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1862 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6,8,9)(7,10,11,12)(13,14)(15,24)(16,20)(17,22)(18,23)(19,25)(21,27)(26,28);(6,9)(7,11)(13,15,17,25,18,16,27,26)(14,19,21,24,23,28,22,20);(6,9)(10,12)(13,16)(14,20)(15,18)(17,26)(19,21)(22,28)(23,24)(25,27);(2,4)(5,7,8,11)(6,10,9,12)(13,15)(14,20)(16,18)(17,26)(19,22)(21,28)(23,24)(25,27)Polycyclic presentation:
2^9|a^2=f,b^2=ef,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,g^b=gi,g^c=gi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 337171);