Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393989 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#127 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#392 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1988 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)(7,15)(8,12)(9,21)(10,14)(11,17)(13,23)(16,31)(18,29)(19,27)(20,35)(22,25)(24,36)(26,33)(28,34)(30,32);(5,7)(6,11)(8,16)(9,22)(10,18)(12,26)(13,29)(14,25)(15,21)(17,23)(19,32)(20,36)(24,27)(28,30)(31,35)(33,34);(5,8)(6,12)(7,16)(9,20)(10,19)(11,24)(13,28)(14,27)(15,30)(17,33)(18,32)(21,35)(22,36)(23,34)(25,26)(29,31);(1,2,3,4)(8,19)(12,27)(16,32)(20,34)(24,26)(28,35)(30,31)(33,36)Polycyclic presentation:
2^9|d^2=h,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72257064, #72257075, #72257079, … |
| maximal normal subgroup of | #72257064, #72257075, #72257079, … |
| maximal quotient of | #72257389, #72257465 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 318903);