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 | |
2340 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)(16,22)(17,24)(18,23)(19,25)(21,30)(26,34)(27,36)(28,35)(29,37)(31,39)(32,38)(33,40)(41,44)(42,43);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15,19,29)(14,20,25,37)(16,26,31,41)(17,27,18,28)(21,33,38,43)(22,34,39,44)(23,35,24,36)(30,40,32,42);(5,7)(6,9)(8,12)(10,11)(13,16,19,31)(14,21,25,38)(15,26,29,41)(17,30,18,32)(20,33,37,43)(22,24,39,23)(27,40,28,42)(34,36,44,35);(1,3,4,2)(7,12)(9,11)(13,16)(14,22)(15,26)(17,32)(18,30)(19,31)(20,34)(21,23)(24,38)(25,39)(27,42)(28,40)(29,41)(33,35)(36,43)(37,44);(6,10)(9,11)(13,17,19,18)(14,23,25,24)(15,27,29,28)(16,30,31,32)(20,35,37,36)(21,22,38,39)(26,40,41,42)(33,34,43,44)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ehi,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,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, 6903752);