Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573591 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3143 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)(3,4)(13,14,19,23)(15,25,28,35)(16,30,33,38)(17,24,18,22)(20,29,36,27)(21,32,37,31)(26,40,42,34)(39,43,44,41);(2,4)(5,6,8,9)(7,10,11,12)(13,15)(14,20)(16,26)(17,29)(18,27)(19,28)(21,34)(22,35)(23,36)(24,25)(30,44)(31,43)(32,41)(33,42)(37,40)(38,39);(6,9)(7,11)(13,16,19,33)(14,21,23,37)(15,26,28,42)(17,31,18,32)(20,34,36,40)(22,30,24,38)(25,39,35,44)(27,41,29,43);(5,7)(6,10)(8,11)(9,12)(13,17,19,18)(14,22,23,24)(15,27,28,29)(16,31,33,32)(20,25,36,35)(21,30,37,38)(26,41,42,43)(34,39,40,44);(6,9)(7,11)(14,23)(15,28)(21,37)(22,24)(26,42)(27,29)(30,38)(41,43)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=i,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=ehi,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=eh83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6502962);