Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437753 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3011 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,10)(8,9)(11,12)(13,14,18,22)(15,23,26,37)(16,20,28,34)(17,30,24,21)(19,31,33,27)(25,36,39,44)(29,41,38,35)(32,42,43,40);(1,2)(3,4)(13,15,24,31,18,26,17,27)(14,19,30,37,22,33,21,23)(16,25,38,42,28,39,29,40)(20,32,41,44,34,43,35,36);(2,4)(6,9)(10,12)(13,16)(14,20)(15,25)(17,29)(18,28)(19,32)(21,35)(22,34)(23,36)(24,38)(26,39)(27,40)(30,41)(31,42)(33,43)(37,44);(2,4)(16,28)(20,34)(25,39)(29,38)(32,43)(35,41)(36,44)(40,42);(5,7,8,11)(6,10,9,12)(15,26)(19,33)(23,37)(25,39)(27,31)(32,43)(36,44)(40,42)Polycyclic presentation:
2^9|a^2=i,b^2=fi,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^b=ei,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6361772);