Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499792 |
| 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 | |
3180 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)(13,14,18,22)(15,23,25,38)(16,20,28,36)(17,30,32,21)(19,31,34,26)(24,39,27,41)(29,44,43,37)(33,42,35,40);(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,27)(17,31)(18,25)(20,35)(21,38)(22,34)(23,30)(24,28)(26,32)(29,40)(33,36)(37,39)(41,44)(42,43);(2,4)(6,9)(7,11)(13,16)(14,20)(15,24)(17,29)(18,28)(19,33)(21,37)(22,36)(23,39)(25,27)(26,40)(30,44)(31,42)(32,43)(34,35)(38,41);(5,7,8,11)(6,10,9,12)(16,28)(20,36)(24,27)(29,43)(33,35)(37,44)(39,41)(40,42);(6,9)(10,12)(15,25)(16,28)(19,34)(20,36)(23,38)(26,31)(29,43)(37,44)Polycyclic presentation:
2^9|a^2=i,d^2=hi,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=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6419852);