Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | #24438 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1830 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)(5,6)(7,9)(8,10)(11,12)(13,14,19,24,20,25,18,35)(15,26,28,38,16,22,30,21)(17,31,33,41,34,23,36,42)(27,39,32,37,29,40,43,44);(5,6)(7,11)(8,10)(9,12)(13,15,20,16)(14,21,25,38)(17,32,34,43)(18,28,19,30)(22,35,26,24)(23,40,31,39)(27,36,29,33)(37,41,44,42);(2,4)(5,7,8,12)(6,9,10,11)(13,16,20,15)(14,22,25,26)(17,29,34,27)(18,28,19,30)(21,35,38,24)(23,39,31,40)(32,33,43,36)(37,41,44,42);(7,12)(9,11)(13,17,18,36,20,34,19,33)(14,23,35,41,25,31,24,42)(15,27,30,43,16,29,28,32)(21,37,22,39,38,44,26,40);(6,10)(9,11)(13,18,20,19)(14,24,25,35)(15,28,16,30)(17,33,34,36)(21,22,38,26)(23,41,31,42)(27,43,29,32)(37,40,44,39)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=hi,d^2=fi,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6972378);