Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438017 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3010 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,9)(8,10)(11,12)(13,14,18,22)(15,23,24,38)(16,20,28,36)(17,30,31,21)(19,32,33,26)(25,39,27,42)(29,44,40,37)(34,43,35,41);(5,7)(6,9)(8,11)(10,12)(13,15,17,26,18,24,31,32)(14,19,21,23,22,33,30,38)(16,27,29,43,28,25,40,41)(20,35,37,42,36,34,44,39);(1,2,3,4)(6,10)(7,11)(15,24)(19,33)(23,38)(25,27)(26,32)(34,35)(39,42)(41,43);(7,11)(9,12)(13,16)(14,20)(15,25)(17,29)(18,28)(19,34)(21,37)(22,36)(23,39)(24,27)(26,41)(30,44)(31,40)(32,43)(33,35)(38,42);(16,28)(20,36)(25,27)(29,40)(34,35)(37,44)(39,42)(41,43)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=ghi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6366870);