Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394026 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | #41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1326 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6)(7,12)(8,10)(9,11)(13,14,17,20,18,22,16,21)(15,19,23,28,25,26,24,27);(5,7,9,8)(6,10,11,12)(13,15)(14,19)(16,24)(17,23)(18,25)(20,28)(21,27)(22,26);(5,8,9,7)(6,10,11,12)(14,20)(15,23)(16,17)(19,26)(21,22)(24,25);(2,4)Polycyclic presentation:
2^9|a^2=fi,b^2=e,c^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,f^c=fi,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72322841 |
| maximal normal subgroup of | #72322841 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 303032);