Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417674 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2409 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)(9,23)(10,15)(11,16)(12,20)(14,22)(18,29)(19,28)(21,27)(24,33)(25,32)(26,30)(31,35)(34,36);(1,2)(3,4)(5,7)(6,12)(8,18)(9,24)(10,20)(11,21)(13,17)(14,31)(15,27)(16,29)(19,25)(22,34)(23,36)(26,35)(28,30)(32,33);(5,8,11,10)(6,13,16,15)(7,18,21,20)(9,22,26,25)(12,27,29,17)(14,30,32,23)(19,24,34,35)(28,31,36,33);(2,4)(5,9,11,26)(6,14,16,32)(7,19,21,34)(8,22,10,25)(12,28,29,36)(13,30,15,23)(17,33,27,31)(18,24,20,35)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,d^b=de,e^b=ei,f^b=fi,e^d=ei,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72238736, #72256170, #72256186, … |
| maximal normal subgroup of | #72238736, #72256170, #72256186, … |
| maximal quotient of | #72238746 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324873);