Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33557 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator | C2×D4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C2×#3551 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2449 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,15,18,28)(8,12)(9,21,23,13)(10,14)(11,17)(16,29,30,36)(19,26)(20,33,32,27)(22,25)(24,31)(34,35);(5,7,10,18)(6,11,14,25)(8,16,19,30)(9,22,23,17)(12,24,26,35)(13,28,21,15)(20,34,32,31)(27,36,33,29);(2,4)(8,19)(12,26)(13,21)(15,28)(16,30)(20,32)(24,35)(31,34);(5,8,10,19)(6,12,14,26)(7,16,18,30)(9,20,23,32)(11,24,25,35)(13,27,21,33)(15,29,28,36)(17,31,22,34)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=i,e^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=ei,f^b=fhi,g^b=ghi,d^c=di,e^c=eh,f^e=fi,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72261638, #72261661, #72261670, … |
| maximal normal subgroup of | #72261638, #72261661, #72261670, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400829);