Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419852 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23400 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1610 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,15,11,12,10,16)(7,17,18,26,21,29,20,13)(8,22,23,27,25,30,24,14)(19,28,33,36,34,35,32,31);(1,2)(3,4)(6,12)(7,18)(9,10)(13,17)(19,32)(20,21)(22,30)(23,24)(26,29)(28,36)(31,35)(33,34);(2,4)(5,7,11,21)(6,13,12,26)(8,19,25,34)(9,20,10,18)(14,28,27,35)(15,29,16,17)(22,31,30,36)(23,32,24,33);(5,8)(6,14,12,27)(7,19)(9,24)(10,23)(11,25)(13,28,26,35)(15,30,16,22)(17,31,29,36)(18,32)(20,33)(21,34)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,f^d=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72255658, #72256282, #72256289 |
| maximal normal subgroup of | #72255658, #72256282, #72256289 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327049);