Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394113 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3961 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)(7,17)(8,13)(9,15)(10,14)(11,16)(12,19)(18,29)(20,27)(21,35)(22,31)(23,30)(24,32)(25,28)(26,33)(34,36);(5,7)(6,12)(8,21)(9,25)(10,19)(11,20)(13,29)(14,27)(15,17)(16,28)(18,23)(22,34)(24,33)(26,30)(31,35)(32,36);(2,4)(5,8)(6,13)(7,18)(9,22)(10,23)(11,24)(12,26)(14,30)(15,31)(16,32)(17,29)(19,33)(20,34)(21,25)(27,36)(28,35);(5,7)(6,12)(8,18)(9,19)(10,25)(11,20)(13,26)(14,17)(15,27)(16,28)(21,23)(22,33)(24,34)(29,30)(31,36)(32,35)Polycyclic presentation:
2^9|e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,g^b=gi,g^c=gi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72252204 |
| maximal normal subgroup of | #72252204 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301309);