Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996375 |
| 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×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1819 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,14,20,33,19,23,18,24)(15,25,17,21,28,38,27,36)(16,29,34,37,31,22,32,44)(26,40,30,43,41,39,42,35);(5,6)(7,11)(8,10)(9,12)(13,15,19,28)(14,21,23,36)(16,26,31,41)(17,20,27,18)(22,35,29,43)(24,38,33,25)(30,34,42,32)(37,40,44,39);(2,4)(5,7,8,12)(6,9,10,11)(13,16)(14,22)(15,26)(17,30)(18,32)(19,31)(20,34)(21,35)(23,29)(24,37)(25,39)(27,42)(28,41)(33,44)(36,43)(38,40);(7,12)(9,11)(13,17,18,15,19,27,20,28)(14,21,24,25,23,36,33,38)(16,30,32,26,31,42,34,41)(22,35,37,39,29,43,44,40);(6,10)(9,11)(14,23)(16,31)(21,36)(24,33)(25,38)(26,41)(30,42)(32,34)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=h,d^2=f,f^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6972083);