Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417859 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2433 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2003 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)(7,17)(8,13)(9,25)(10,27)(11,16)(12,20)(14,26)(15,18)(19,33)(21,36)(22,31)(23,32)(24,30)(28,35)(29,34);(5,7,18,26,11,9,10,20)(6,12,27,25,16,14,15,17)(8,19,24,35,23,21,22,34)(13,29,31,36,32,28,30,33);(1,2,3,4)(5,8)(6,13)(7,19)(9,21)(10,24)(11,23)(12,28)(14,29)(15,31)(16,32)(17,33)(18,22)(20,35)(25,36)(26,34)(27,30);(5,9,11,7)(6,14,16,12)(8,21,23,19)(10,20,18,26)(13,29,32,28)(15,17,27,25)(22,34,24,35)(30,36,31,33)Polycyclic presentation:
2^9|b^2=ei,c^2=f,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,d^b=de,g^b=gi,e^c=ei,e^d=ei,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72255729, #72262411, #72262421, … |
| maximal normal subgroup of | #72255729, #72262411, #72262421, … |
| maximal quotient of | #72314550 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330118);