Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #2388 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1843 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,9)(8,10)(11,12)(13,14,19,22,18,24,17,23)(15,25,28,20,26,40,29,35)(16,30,33,38,31,21,34,39)(27,42,44,37,32,41,43,36);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(15,26)(16,31)(20,35)(21,30)(25,40)(28,29)(33,34)(38,39);(5,7,8,12)(6,9,10,11)(13,15)(14,20)(16,32)(17,28)(18,26)(19,29)(21,37)(22,25)(23,40)(24,35)(27,31)(30,36)(33,44)(34,43)(38,41)(39,42);(2,4)(7,12)(9,11)(13,16)(14,21)(15,27)(17,34)(18,31)(19,33)(20,36)(22,39)(23,38)(24,30)(25,41)(26,32)(28,44)(29,43)(35,37)(40,42);(6,10)(9,11)(13,17,18,19)(14,22,24,23)(15,28,26,29)(16,33,31,34)(20,25,35,40)(21,38,30,39)(27,43,32,44)(36,42,37,41)Polycyclic presentation:
2^9|a^2=gi,c^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfi,e^b=ehi,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6969760);