Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| 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 | C2×#550 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1846 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,10)(8,11)(9,12)(13,14,19,20,18,24,17,23)(15,21,31,33,26,36,27,37)(16,28,32,38,30,22,29,34)(25,39,42,44,41,35,40,43);(5,6)(7,10)(8,11)(9,12)(14,20)(16,29)(17,19)(21,33)(23,24)(25,40)(27,31)(30,32)(34,38)(36,37)(41,42)(43,44);(1,2,3,4)(5,7,10,6)(8,11,9,12)(13,15,18,26)(14,21,24,36)(16,25,30,41)(17,27,19,31)(20,33,23,37)(22,35,28,39)(29,40,32,42)(34,43,38,44);(5,8,10,9)(6,11,7,12)(13,16,17,29,18,30,19,32)(14,22,23,38,24,28,20,34)(15,25,27,40,26,41,31,42)(21,35,37,44,36,39,33,43);(5,9,10,8)(6,11,7,12)(15,26)(21,36)(25,41)(27,31)(33,37)(35,39)(40,42)(43,44)Polycyclic presentation:
2^9|a^2=fi,c^2=gi,d^2=fh,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7106159);