Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13318 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1999 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C2×#3067 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1028 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(6,12,14,27)(8,21)(9,10)(13,29,30,26)(15,28,16,22)(17,32,31,35)(18,33)(19,25)(23,24)(34,36);(5,6)(7,17)(8,22)(9,16)(10,15)(11,14)(12,23)(13,19)(18,26)(20,31)(21,27)(24,28)(25,30)(29,34)(32,36)(33,35);(2,4)(5,7)(6,13)(8,18)(9,25)(10,19)(11,20)(12,26)(14,30)(15,31)(16,17)(21,33)(22,35)(23,36)(24,34)(27,29)(28,32)Polycyclic presentation:
2^9|a^2=g,d^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,d^b=dhi,e^b=ei,f^b=fi,g^b=gi,h^b=hi,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72249868, #72249869, #72249885, … |
| maximal normal subgroup of | #72249868, #72249869, #72249885, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54880);