Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417541 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2404 |
| prime radical | C2×#3018 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2019 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)(7,16,19,30)(8,13)(9,14)(10,23)(11,15)(12,20,26,17)(18,31,33,36)(21,27)(22,28)(24,29)(25,34,35,32);(7,17,19,20)(8,21)(12,16,26,30)(13,27)(18,32,33,34)(22,24)(25,31,35,36)(28,29);(5,7)(6,12)(8,18)(9,17)(10,20)(11,19)(13,25)(14,30)(15,26)(16,23)(21,33)(22,32)(24,34)(27,35)(28,36)(29,31);(2,4)(5,8)(6,13)(7,18)(9,22)(10,24)(11,21)(12,25)(14,28)(15,27)(16,31)(17,32)(19,33)(20,34)(23,29)(26,35)(30,36)Polycyclic presentation:
2^9|a^2=e,b^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=dei,e^c=ei,f^c=fi,g^c=gi,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72263012, C2×#417959, #72263055, … |
| maximal normal subgroup of | #72263012, C2×#417959, #72263055, … |
| maximal quotient of | C2×#417959 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325155);