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)(13,14,19,23)(15,24,29,40)(16,21,31,39)(17,22,18,33)(20,27,36,28)(25,41,26,42)(30,37,32,38)(34,44,35,43);(5,6)(7,11)(8,10)(9,12)(13,15)(14,20)(16,26)(17,28)(18,27)(19,29)(21,35)(22,40)(23,36)(24,33)(25,31)(30,43)(32,44)(34,39)(37,42)(38,41);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16,19,31)(14,21,23,39)(15,25,29,26)(17,32,18,30)(20,34,36,35)(22,38,33,37)(24,41,40,42)(27,44,28,43);(7,12)(9,11)(13,16)(14,21)(15,26)(17,32)(18,30)(19,31)(20,35)(22,38)(23,39)(24,42)(25,29)(27,43)(28,44)(33,37)(34,36)(40,41);(6,10)(9,11)(13,17,19,18)(14,22,23,33)(15,27,29,28)(16,30,31,32)(20,24,36,40)(21,37,39,38)(25,43,26,44)(34,42,35,41)Polycyclic presentation:
2^9|a^2=i,c^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, 6407085);