Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498904 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C2×#3021 | C2 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
2161 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)(5,6,11,16)(7,17,21,31)(8,15,10,13)(9,23,25,14)(12,24,29,20)(18,33,19,34)(22,30,26,32)(27,35,28,36);(5,7)(6,12)(8,19)(9,24)(10,18)(11,21)(13,28)(14,31)(15,27)(16,29)(17,23)(20,25)(22,35)(26,36)(30,34)(32,33);(2,4)(7,18)(8,10)(12,27)(13,15)(17,33)(19,21)(20,36)(22,26)(24,35)(28,29)(30,32)(31,34);(5,8,11,10)(6,13,16,15)(7,19,21,18)(9,22,25,26)(12,28,29,27)(14,30,23,32)(17,33,31,34)(20,36,24,35)Polycyclic presentation:
2^9|a^2=i,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,g^a=ghi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,d^c=di,e^c=eh,g^c=gi,g^e=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72312685, #72312776 |
| maximal normal subgroup of | #72312685, #72312776 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406106);