Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417575 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1200 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)(5,6,9,11)(7,12,8,10)(13,14,17,20)(15,22,25,18)(16,21,27,19)(23,26,28,24);(13,14,16,19,17,20,27,21)(15,23,24,22,25,28,26,18);(5,7,9,8)(6,10,11,12)(13,15,17,25)(14,18,20,22)(16,24,27,26)(19,23,21,28);(2,4)(5,8,9,7)(6,10,11,12)(13,14)(15,18)(16,21)(17,20)(19,27)(22,25)(23,26)(24,28)Polycyclic presentation:
2^9|a^2=f,b^2=e,c^2=f,d^2=fi,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, 338212);