Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394098 |
| 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 | |
4021 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)(7,19)(8,14)(9,25)(10,18)(11,16)(12,17)(13,22)(15,26)(20,34)(21,29)(23,33)(24,32)(27,30)(28,36)(31,35);(5,7)(6,13)(8,20)(9,21)(10,27)(11,22)(12,23)(14,31)(15,29)(16,33)(17,30)(18,19)(24,36)(25,28)(26,35)(32,34);(2,4)(5,8,12,9)(6,14,17,25)(7,20,23,21)(10,26,11,24)(13,28,30,31)(15,16,32,18)(19,34,33,29)(22,36,27,35);(5,9,12,8)(6,15,17,32)(7,21,23,20)(10,26,11,24)(13,29,30,34)(14,16,25,18)(19,31,33,28)(22,36,27,35)Polycyclic presentation:
2^9|c^2=i,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,e^c=ei,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72236660, #72261238, #72303145, … |
| maximal normal subgroup of | #72236660, #72261238, #72303145, … |
| maximal quotient of | #72236668, #72262065 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301297);