Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213778 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2550 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)(7,15,18,11)(8,19)(9,23)(10,13)(12,22)(14,25)(16,31,20,33)(17,27,24,29)(21,34)(26,32,30,35)(28,36);(2,4)(5,7)(6,11)(8,20)(9,17)(10,18)(12,26)(13,15)(14,27)(16,21)(19,33)(22,35)(23,29)(24,25)(28,30)(31,34)(32,36);(5,8,9,22,10,21,25,36)(6,12,14,19,13,28,23,34)(7,16,24,35,18,20,17,32)(11,26,29,31,15,30,27,33);(2,4)(5,7)(6,11)(8,16)(9,24)(10,18)(12,26)(13,15)(14,29)(17,25)(19,31)(20,21)(22,35)(23,27)(28,30)(32,36)(33,34)Polycyclic presentation:
2^9|a^2=h,c^2=fh,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,h^b=hi,e^c=ei,g^c=gi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281228);