Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432861 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3500 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4292 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)(5,6)(7,9)(8,10)(11,12)(13,14,19,24)(15,25,29,33)(16,22,30,37)(17,23,18,21)(20,28,35,27)(26,39,40,43)(31,38,32,36)(34,42,44,41);(2,4)(5,7)(6,9)(8,11)(10,12)(13,15,19,29)(14,20,24,35)(16,26,30,40)(17,28,18,27)(21,33,23,25)(22,34,37,44)(31,42,32,41)(36,43,38,39);(7,11)(9,12)(14,21)(17,18)(20,33)(22,36)(23,24)(25,35)(27,28)(31,32)(34,43)(37,38)(39,44)(41,42);(6,10)(9,12)(13,16)(14,22)(15,26)(17,31)(18,32)(19,30)(20,34)(21,36)(23,38)(24,37)(25,39)(27,41)(28,42)(29,40)(33,43)(35,44);(16,30)(22,37)(26,40)(31,32)(34,44)(36,38)(39,43)(41,42)Polycyclic presentation:
2^9|a^2=i,b^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,c^b=ch,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6340134);