Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500076 |
| 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×#3552 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3014 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)(13,14,19,33,18,22,17,23)(15,24,31,20,26,37,27,35)(16,28,32,21,30,38,29,36)(25,34,42,39,40,44,41,43);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,25)(17,31)(18,26)(19,27)(21,34)(22,35)(23,37)(24,33)(28,39)(29,42)(30,40)(32,41)(36,44)(38,43);(6,9)(7,11)(13,16,18,30)(14,21,22,36)(15,25,26,40)(17,32,19,29)(20,34,35,44)(23,38,33,28)(24,39,37,43)(27,42,31,41);(5,7,8,11)(6,10,9,12)(14,22)(15,26)(21,36)(23,33)(25,40)(27,31)(28,38)(41,42);(6,9)(10,12)Polycyclic presentation:
2^9|a^2=fi,c^2=i,d^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6420398);