Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395586 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2411 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14918 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1799 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,8,12,7,13,6,22)(3,14,24,26,16,31,15,9)(4,10,23,27,19,28,18,30)(5,17,20,29,21,11,25,32);(1,3)(2,9,13,26)(4,17,19,11)(5,10)(6,15)(7,16)(8,24)(12,14,22,31)(18,32,23,29)(20,27)(21,28)(25,30);(1,4)(2,10)(3,11)(5,9)(6,23)(7,19)(8,18)(12,30)(13,28)(14,25)(15,32)(16,17)(20,31)(21,26)(22,27)(24,29);(1,5)(2,11)(3,10)(4,9)(6,20)(7,21)(8,25)(12,29)(13,17)(14,18)(15,27)(16,28)(19,26)(22,32)(23,31)(24,30)Polycyclic presentation:
2^9|a^2=ef,b^2=h,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,g^b=gi,e^c=ei,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72338517, #72338521, #72338526 |
| maximal normal subgroup of | #72338517, #72338521, #72338526 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 312964);