Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417603 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1133 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6,9,11)(7,10,8,12)(13,14)(15,23)(16,24)(17,20)(18,22)(19,26)(21,28)(25,27);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,18,25)(14,16,22,26)(17,19,28,24)(20,23,21,27);(5,8,9,7)(6,10,11,12)(13,16,18,26)(14,19,22,24)(15,21,25,20)(17,27,28,23);(2,4)(14,20)(15,19)(17,28)(21,22)(23,27)(24,25)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,e^c=ei,g^c=gi,e^d=ei,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 333758);