Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455699 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2797 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16268 |
| prime radical | #23412 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1092 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,15)(4,18)(5,21)(6,14)(7,13)(8,12)(9,10)(11,25)(16,26)(17,20)(19,31)(22,30)(23,29)(24,28)(27,32);(1,3,8,18,7,17,6,10)(2,9,14,20,13,4,12,15)(5,16,23,30,24,31,25,27)(11,26,28,32,29,19,21,22);(1,4)(2,10)(3,12)(5,22)(6,15)(7,9)(8,20)(11,16)(13,18)(14,17)(19,23)(21,30)(24,32)(25,26)(27,28)(29,31);(1,5)(2,11)(3,16)(4,19)(6,25)(7,24)(8,23)(9,26)(10,27)(12,21)(13,29)(14,28)(15,22)(17,31)(18,30)(20,32)Polycyclic presentation:
2^9|b^2=ef,e^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,g^b=gi,h^b=hi,d^c=de,e^c=ei,e^d=ei,f^d=fi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 362895);