Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| 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 | #30413 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1863 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,8)(9,11)(10,12)(13,14)(15,19)(16,27)(17,20)(18,23)(21,29)(22,31)(24,40)(25,35)(26,33)(28,44)(30,39)(32,38)(34,41)(36,42)(37,43);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15)(14,19)(16,28)(17,26)(18,25)(20,33)(21,37)(22,36)(23,35)(24,30)(27,44)(29,43)(31,42)(32,41)(34,38)(39,40);(2,4)(14,20)(17,31)(19,33)(21,27)(22,23)(26,42)(29,32)(34,40)(35,36)(37,44)(38,39)(41,43);(6,11)(7,12)(13,16,18,30)(14,21,23,38)(15,24,25,28)(17,32,31,29)(19,34,35,37)(20,27,22,39)(26,43,42,41)(33,40,36,44);(5,8,9,10)(6,7,11,12)(15,25)(19,35)(24,28)(26,42)(33,36)(34,37)(40,44)(41,43)Polycyclic presentation:
2^9|d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=dfi,e^b=ei,g^c=gi,g^d=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410149);