Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394353 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2553 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)(5,6,11,16)(7,12,19,27)(8,20,22,30)(9,15,10,14)(13,24,28,21)(17,29,23,25)(18,32,33,35)(26,31,36,34);(7,17,19,23)(12,25,27,29)(18,31,33,34)(26,35,36,32);(2,4)(5,7)(6,12)(8,18)(9,23)(10,17)(11,19)(13,26)(14,29)(15,25)(16,27)(20,32)(21,34)(22,33)(24,31)(28,36)(30,35);(5,8)(6,13)(7,18)(9,21)(10,24)(11,22)(12,26)(14,20)(15,30)(16,28)(17,31)(19,33)(23,34)(25,35)(27,36)(29,32)Polycyclic presentation:
2^9|a^2=i,b^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,e^c=ei,g^c=gi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72341337, #72341353, #72341354 |
| maximal normal subgroup of | #72341337, #72341353, #72341354 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301551);