Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438991 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2038 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,4,3)(5,6)(7,11)(8,12)(9,10)(13,14)(15,24)(16,27)(17,20)(18,22)(19,33)(21,30)(23,32)(25,39)(26,35)(28,41)(29,42)(31,38)(34,43)(36,40)(37,44);(2,3)(6,10)(11,12)(13,15)(14,20)(16,28)(17,18)(19,26)(21,36)(22,24)(23,35)(25,29)(27,39)(30,43)(31,32)(33,38)(34,37)(40,44)(41,42);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,16,19,30,18,29,32,44)(14,21,23,27,22,37,33,42)(15,25,26,40,17,28,31,43)(20,34,35,41,24,36,38,39);(1,3,4,2)(5,8,9,7)(6,12,10,11)(13,17,18,15)(14,20,22,24)(16,28,29,25)(19,31,32,26)(21,34,37,36)(23,35,33,38)(27,41,42,39)(30,43,44,40);(15,17)(16,29)(20,24)(21,37)(26,31)(27,42)(30,44)(35,38)Polycyclic presentation:
2^9|a^2=hi,c^2=fgi,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,e^c=ei,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6364825);