Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438997 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2024 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,4,3)(5,6)(7,11)(8,12)(9,10)(13,14)(15,24)(16,27)(17,20)(18,22)(19,33)(21,31)(23,28)(25,39)(26,35)(29,41)(30,42)(32,38)(34,44)(36,43)(37,40);(2,3)(6,10)(11,12)(13,15)(14,20)(16,25)(17,18)(19,26)(21,34)(22,24)(23,35)(27,41)(28,32)(29,30)(31,40)(33,38)(36,37)(39,42)(43,44);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,16,28,43,18,30,19,31)(14,21,33,42,22,36,23,27)(15,25,32,44,17,29,26,40)(20,34,38,39,24,37,35,41);(1,3,4,2)(5,8,9,7)(6,12,10,11)(13,17,18,15)(14,20,22,24)(16,29,30,25)(19,32,28,26)(21,34,36,37)(23,35,33,38)(27,41,42,39)(31,44,43,40);(15,17)(16,30)(20,24)(21,36)(26,32)(27,42)(31,43)(35,38)Polycyclic presentation:
2^9|a^2=h,c^2=fg,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6364830);