Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394353 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3068 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1894 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,13)(8,22)(9,16)(10,15)(11,17)(12,19)(14,23)(18,32)(20,28)(21,27)(24,30)(25,31)(26,35)(29,33)(34,36);(5,7,10,20,11,21,9,19)(6,12,16,27,17,28,15,13)(8,18,25,34,24,35,23,33)(14,26,30,36,31,32,22,29);(1,2,3,4)(5,7,11,21)(6,13,17,27)(8,18,24,35)(9,19,10,20)(12,15,28,16)(14,29,31,36)(22,32,30,26)(23,33,25,34);(5,8)(6,14)(7,18)(9,23)(10,25)(11,24)(12,26)(13,29)(15,22)(16,30)(17,31)(19,33)(20,34)(21,35)(27,36)(28,32)Polycyclic presentation:
2^9|b^2=ei,c^2=fi,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,g^b=gi,g^c=gi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72317311, #72319703, #72319923, … |
| maximal normal subgroup of | #72317311, #72319703, #72319923, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 305161);