Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #137659 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1893 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
652 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,9,16,11,14,10,27)(7,17,19,31,20,12,26,30)(8,21,24,33,25,13,23,15)(18,28,35,34,22,32,36,29);(1,2)(3,4)(5,7)(6,12)(8,22)(9,26)(10,19)(11,20)(13,32)(14,17)(15,34)(16,31)(18,25)(21,28)(23,36)(24,35)(27,30)(29,33);(2,4)(6,13,14,21)(8,23)(9,10)(12,28,17,32)(15,16,33,27)(18,35)(19,26)(22,36)(24,25)(29,31,34,30)Polycyclic presentation:
2^9|a^2=g,c^2=d,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fi,h^a=hi,c^b=cf,e^b=ei,g^b=gi,e^c=eh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72249868, #72249873, #72249879, … |
| maximal normal subgroup of | #72249868, #72249873, #72249879, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55802);