Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2042 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,14,18,23,19,24,34,41)(15,25,29,37,28,20,27,38)(16,22,31,39,17,32,30,21)(26,36,43,35,33,40,44,42);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15)(14,20)(16,26)(17,33)(18,29)(19,28)(21,35)(22,40)(23,38)(24,25)(27,34)(30,44)(31,43)(32,36)(37,41)(39,42);(5,7)(6,9)(8,12)(10,11)(13,16,19,17)(14,21,24,39)(15,26,28,33)(18,30,34,31)(20,35,25,42)(22,23,32,41)(27,43,29,44)(36,37,40,38);(1,3,4,2)(7,12)(9,11)(13,17,19,16)(14,22,24,32)(15,26,28,33)(18,30,34,31)(20,36,25,40)(21,23,39,41)(27,44,29,43)(35,37,42,38);(6,10)(9,11)(13,18,19,34)(14,23,24,41)(15,27,28,29)(16,30,17,31)(20,37,25,38)(21,32,39,22)(26,43,33,44)(35,40,42,36)Polycyclic presentation:
2^9|a^2=fgi,c^2=i,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=ehi,d^c=dh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6913736);