Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #422540 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2669 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23399 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1234 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,19,23,32)(8,24,25,36)(9,27)(10,16)(11,17)(12,18)(13,22,31,30)(14,21,33,26)(15,20)(28,35)(29,34);(1,2)(3,4)(5,7,11,22,12,23,10,30)(6,13,17,19,18,31,16,32)(8,20,21,9,25,28,26,29)(14,15,24,27,33,35,36,34);(2,4)(5,8,10,21,12,25,11,26)(6,14,16,24,18,33,17,36)(7,20,22,29,23,28,30,9)(13,15,19,34,31,35,32,27);(5,9,10,20,12,29,11,28)(6,15,16,34,18,35,17,27)(7,21,22,25,23,26,30,8)(13,14,19,24,31,33,32,36)Polycyclic presentation:
2^9|a^2=ef,b^2=f,c^2=e,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,f^c=fi,g^c=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 329736);