Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23872 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3499 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3746 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)(7,19)(8,24)(9,27)(10,17)(11,16)(12,18)(13,21)(14,25)(15,28)(20,33)(22,32)(23,31)(26,36)(29,35)(30,34);(5,7,12,23)(6,13,18,32)(8,20,26,9)(10,22,11,21)(14,33,34,27)(15,24,35,36)(16,31,17,19)(25,28,30,29);(5,8)(6,14)(7,20)(9,23)(10,25)(11,30)(12,26)(13,27)(15,31)(16,24)(17,36)(18,34)(19,35)(21,29)(22,28)(32,33);(5,9,12,20)(6,15,18,35)(7,8,23,26)(10,29,11,28)(13,24,32,36)(14,31,34,19)(16,33,17,27)(21,30,22,25)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,f^c=fi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72261238, #72261282, #72306512, … |
| maximal normal subgroup of | #72261238, #72261282, #72306512, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266868);