Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454095 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2310 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)(7,16)(8,12)(9,13)(10,15)(11,18)(14,25)(17,33)(19,28)(20,26)(21,30)(22,31)(23,32)(24,27)(29,35)(34,36);(5,7)(6,11)(8,20)(9,24)(10,19)(12,29)(13,16)(14,28)(15,27)(17,22)(18,25)(21,35)(23,34)(26,31)(30,36)(32,33);(5,8)(6,12)(7,17)(9,21)(10,23)(11,26)(13,30)(14,31)(15,32)(16,29)(18,34)(19,35)(20,24)(22,25)(27,36)(28,33);(2,4)(6,13)(8,21)(9,25)(11,16)(14,15)(17,34)(18,24)(20,35)(22,23)(26,36)(27,28)(30,31)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,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72255650, #72262411, #72262422, … |
| maximal normal subgroup of | #72255650, #72262411, #72262422, … |
| maximal quotient of | #72314563 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361291);