Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500186 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3551 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3182 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)(13,14,19,33,18,23,17,22)(15,24,32,20,28,40,27,36)(16,21,29,37,31,39,30,38)(25,41,43,34,26,42,44,35);(5,6)(7,11)(8,10)(9,12)(13,15)(14,20)(16,26)(17,32)(18,28)(19,27)(21,35)(22,40)(23,36)(24,33)(25,31)(29,43)(30,44)(34,39)(37,42)(38,41);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16,18,31)(14,21,23,39)(15,25,28,26)(17,30,19,29)(20,34,36,35)(22,38,33,37)(24,41,40,42)(27,44,32,43);(7,12)(9,11)(13,16)(14,21)(15,26)(17,30)(18,31)(19,29)(20,35)(22,38)(23,39)(24,42)(25,28)(27,43)(32,44)(33,37)(34,36)(40,41);(6,10)(9,11)(13,17,18,19)(14,22,23,33)(15,27,28,32)(16,29,31,30)(20,24,36,40)(21,37,39,38)(25,43,26,44)(34,42,35,41)Polycyclic presentation:
2^9|a^2=fi,c^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,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, 6407418);