Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393921 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2846 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3951 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(6,11)(8,19)(9,21)(12,26)(14,22)(15,31)(16,17)(20,23)(24,28)(25,29)(27,36)(32,33);(1,2)(3,4)(5,6)(7,12)(8,13)(9,22)(10,14)(11,21)(15,27)(16,24)(17,26)(18,28)(19,25)(20,30)(23,29)(31,34)(32,35)(33,36);(2,4)(5,7)(6,12)(8,15)(9,17)(10,18)(11,24)(13,27)(14,28)(16,21)(19,31)(20,33)(22,26)(23,32)(25,34)(29,35)(30,36);(5,8,10,20)(6,13,14,30)(7,15,18,33)(9,23,21,19)(11,25,22,29)(12,27,28,36)(16,31,17,32)(24,34,26,35)Polycyclic presentation:
2^9|d^2=i,e^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,h^b=hi,e^c=ei,e^d=ei,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72249921, #72251058, #72251089, … |
| maximal normal subgroup of | #72249921, #72251058, #72251089, … |
| maximal quotient of | #72249963, #72249965, #72257614, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301118);