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 | |
2321 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)(13,14,18,24)(15,25,29,20)(16,22,32,38)(17,23,19,34)(21,33,39,31)(26,36,27,40)(28,42,30,37)(35,43,41,44);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15)(14,20)(16,26)(17,28)(18,29)(19,30)(21,35)(22,40)(23,37)(24,25)(27,32)(31,43)(33,44)(34,42)(36,38)(39,41);(5,7)(6,9)(8,12)(10,11)(13,16)(14,21)(15,26)(17,31)(18,32)(19,33)(20,35)(22,34)(23,38)(24,39)(25,41)(27,29)(28,43)(30,44)(36,37)(40,42);(1,3,4,2)(7,12)(9,11)(13,16)(14,22)(15,27)(17,33)(18,32)(19,31)(20,36)(21,34)(23,39)(24,38)(25,40)(26,29)(28,43)(30,44)(35,37)(41,42);(6,10)(9,11)(13,17,18,19)(14,23,24,34)(15,28,29,30)(16,31,32,33)(20,37,25,42)(21,38,39,22)(26,43,27,44)(35,36,41,40)Polycyclic presentation:
2^9|a^2=f,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,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, 6903678);