Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33568 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator | C2×Q8 | C23×C4 |
| Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C2×#3018 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1939 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,8,14,11,17,9,15)(7,18,20,12,22,28,19,30)(10,16,24,13,26,31,23,32)(21,33,34,36,25,27,35,29);(1,2,3,4)(5,7)(6,12)(8,19)(9,20)(10,25)(11,22)(13,27)(14,18)(15,28)(16,29)(17,30)(21,26)(23,35)(24,34)(31,36)(32,33);(5,8,11,9)(6,13,17,32)(7,19,22,20)(10,24,26,23)(12,27,30,33)(14,16,15,31)(18,29,28,36)(21,35,25,34);(5,8,11,9)(6,14,17,15)(7,20,22,19)(10,23,26,24)(12,28,30,18)(13,16,32,31)(21,35,25,34)(27,36,33,29)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=i,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fhi,h^a=hi,d^b=di,e^b=ei,f^b=fhi,g^b=gi,e^c=eh,f^c=fi,g^c=gh,f^e=fi,g^f=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72286473, #72286480, #72286516, … |
| maximal normal subgroup of | #72286473, #72286480, #72286516, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401816);