Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2357 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)(7,9)(8,10)(11,12)(13,14)(15,20)(16,26)(17,22)(18,31)(19,23)(21,24)(25,34)(27,39)(28,33)(29,37)(30,41)(32,36)(35,38)(40,43)(42,44);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15,19,18)(14,20,23,31)(16,24,28,32)(17,25,29,30)(21,33,36,26)(22,34,37,41)(27,38,40,42)(35,43,44,39);(5,7)(6,9)(8,12)(10,11)(13,16,19,28)(14,21,23,36)(15,24,18,32)(17,27,29,40)(20,33,31,26)(22,35,37,44)(25,38,30,42)(34,43,41,39);(1,3,4,2)(7,12)(9,11)(13,17)(14,22)(15,25)(16,27)(18,30)(19,29)(20,34)(21,35)(23,37)(24,38)(26,39)(28,40)(31,41)(32,42)(33,43)(36,44);(6,10)(9,11)(17,29)(22,37)(25,30)(27,40)(34,41)(35,44)(38,42)(39,43)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6903687);