Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #457102 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2428 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1337 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)(5,6,9,14,10,15,24,12)(7,16,17,11,20,27,19,25)(8,21,22,13,23,29,35,28)(18,32,31,30,33,26,34,36);(5,7,10,20)(6,11)(8,18,23,33)(9,17,24,19)(12,16)(13,26)(14,27)(15,25)(21,30)(22,31,35,34)(28,32)(29,36);(6,12)(7,17)(8,22)(9,24)(11,25)(13,28)(14,15)(18,33)(19,20)(23,35)(26,36)(30,32);(2,4)(5,8,9,22,10,23,24,35)(6,13,12,21,15,28,14,29)(7,18,17,31,20,33,19,34)(11,26,16,30,25,32,27,36)Polycyclic presentation:
2^9|a^2=f,b^2=ef,d^2=e,e^2=i,f^2=i,g^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,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 364450);