Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #137655 |
| 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 | C23×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
728 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,11,17,10,16,24,15,8,14,26,13,25,35,9,27)(7,18,23,34,19,31,36,29,20,12,28,32,22,33,21,30);(1,2,3,4)(5,7,8,20)(6,12,14,18)(9,21,24,36)(10,19,25,22)(11,23,26,28)(13,34,17,32)(15,30,27,29)(16,33,35,31);(6,13)(7,19)(9,11)(10,25)(12,29)(14,17)(15,16)(18,30)(20,22)(21,36)(24,26)(27,35)(31,32)(33,34)Polycyclic presentation:
2^9|a^2=g,b^2=d,e^2=hi,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,e^a=ei,c^b=cf,f^b=fi,e^c=eh,f^c=fi,g^c=ghi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239809, #72239813, #72239839, … |
| maximal normal subgroup of | #72239809, #72239813, #72239839, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55395);