Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806623 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1765 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)(7,11)(8,10)(9,12)(13,14,19,24)(15,20,27,34)(16,21,25,32)(17,22,28,35)(18,23,31,38)(26,33,40,43)(29,36,39,42)(30,37,41,44);(5,7,8,12)(6,9,10,11)(13,15)(14,20)(16,28)(17,25)(18,29)(19,27)(21,35)(22,32)(23,36)(24,34)(26,30)(31,39)(33,37)(38,42)(40,41)(43,44);(1,2,3,4)(6,10)(7,12)(13,16,28,15,19,25,17,27)(14,21,35,20,24,32,22,34)(18,29,41,26,31,39,30,40)(23,36,44,33,38,42,37,43);(6,10)(9,11)(13,17,19,28)(14,22,24,35)(15,25,27,16)(18,30,31,41)(20,32,34,21)(23,37,38,44)(26,39,40,29)(33,42,43,36);(13,18,28,41,19,31,17,30)(14,23,35,44,24,38,22,37)(15,26,25,39,27,40,16,29)(20,33,32,42,34,43,21,36)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=ghi,d^2=i,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6723398);