Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394602 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#394 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2163 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,16)(8,21)(9,26)(10,14)(11,15)(12,18)(13,24)(17,32)(19,28)(20,31)(22,27)(23,30)(25,35)(29,34)(33,36);(5,7,11,20)(6,12,15,28)(8,22,24,33)(9,19,10,18)(13,29,21,32)(14,31,26,16)(17,23,34,25)(27,35,36,30);(5,8,9,23,11,24,10,25)(6,13,14,30,15,21,26,35)(7,17,18,33,20,34,19,22)(12,27,16,29,28,36,31,32);(2,4)Polycyclic presentation:
2^9|b^2=i,c^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,g^b=gi,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72285454, #72285470, #72285654 |
| maximal normal subgroup of | #72285454, #72285470, #72285654 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301800);