Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996719 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1805 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(6,10)(8,11)(13,14,18,21)(15,23,24,36)(16,20,27,35)(17,29,30,22)(19,31,32,26)(25,37,38,44)(28,41,40,34)(33,42,43,39);(1,2)(3,4)(5,6,9,10)(7,8,12,11)(15,24)(19,32)(23,36)(25,38)(26,31)(33,43)(37,44)(39,42);(5,7)(6,11)(8,10)(9,12)(13,15,17,26,18,24,30,31)(14,19,22,23,21,32,29,36)(16,25,28,39,27,38,40,42)(20,33,34,37,35,43,41,44);(2,4)(6,10)(7,12)(13,16)(14,20)(15,25)(17,28)(18,27)(19,33)(21,35)(22,34)(23,37)(24,38)(26,39)(29,41)(30,40)(31,42)(32,43)(36,44);(5,8,9,11)(6,7,10,12)(14,21)(16,27)(19,32)(22,29)(23,36)(25,38)(28,40)(39,42)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=g,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7092520);