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 |
| prime radical | C2×#3516 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2952 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)(13,14,17,32,19,23,18,22)(15,24,26,20,27,38,28,35)(16,29,33,21,31,36,30,37)(25,34,42,39,41,44,40,43);(2,4)(5,6,8,9)(7,10,11,12)(13,15)(14,20)(16,25)(17,28)(18,26)(19,27)(21,34)(22,38)(23,35)(24,32)(29,39)(30,42)(31,41)(33,40)(36,43)(37,44);(6,9)(7,11)(13,16,19,31)(14,21,23,37)(15,25,27,41)(17,30,18,33)(20,34,35,44)(22,36,32,29)(24,39,38,43)(26,40,28,42);(5,7)(6,10)(8,11)(9,12)(13,17,19,18)(14,22,23,32)(15,26,27,28)(16,30,31,33)(20,24,35,38)(21,36,37,29)(25,40,41,42)(34,39,44,43);(6,9)(7,11)(14,23)(15,27)(21,37)(22,32)(25,41)(26,28)(29,36)(40,42)Polycyclic presentation:
2^9|a^2=fi,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,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, 6503514);