Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499823 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
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 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3890 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(13,14)(15,26)(16,22)(17,23)(18,24)(19,34)(20,25)(21,29)(27,41)(28,36)(30,40)(31,44)(32,38)(33,39)(35,42)(37,43);(5,6)(7,11)(8,10)(9,12)(13,15)(14,21)(16,31)(17,27)(18,29)(19,28)(20,30)(22,37)(23,35)(24,40)(25,36)(26,34)(32,42)(33,43)(38,44)(39,41);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16)(14,22)(15,27)(17,20)(18,33)(19,32)(21,35)(23,25)(24,39)(26,41)(28,43)(29,42)(30,31)(34,38)(36,37)(40,44);(7,12)(9,11)(13,17,20,16)(14,23,25,22)(15,27,30,31)(18,32,19,33)(21,35,36,37)(24,38,34,39)(26,41,40,44)(28,43,29,42);(6,10)(9,11)(13,18,20,19)(14,24,25,34)(15,28,30,29)(16,32,17,33)(21,26,36,40)(22,38,23,39)(27,42,31,43)(35,44,37,41)Polycyclic presentation:
2^9|d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6407079);