Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417530 |
| 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 | |
3344 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,12,18,27)(8,13)(9,11)(10,15)(14,21)(16,30,32,25)(17,23,33,26)(19,29)(20,28)(22,24)(31,35,34,36);(1,2)(3,4)(6,11)(8,19)(9,21)(12,23)(13,28)(14,15)(16,31)(17,33)(20,22)(25,30)(26,27)(32,34);(2,4)(5,7)(6,12)(8,16)(9,17)(10,18)(11,23)(13,25)(14,26)(15,27)(19,31)(20,32)(21,33)(22,34)(24,35)(28,30)(29,36);(5,8)(6,13)(7,16)(9,22)(10,20)(11,24)(12,25)(14,29)(15,28)(17,34)(18,32)(19,21)(23,35)(26,36)(27,30)(31,33)Polycyclic presentation:
2^9|a^2=h,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,h^c=hi,e^d=ei,f^d=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72239541, #72255954, #72256189, … |
| maximal normal subgroup of | #72239541, #72255954, #72256189, … |
| maximal quotient of | #72239555, C2×#449495, #72267314, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356691);