Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518964 |
| 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 | |
1843 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)(13,14,20,24)(15,21,27,38)(16,28,26,37)(17,30,34,44)(18,23,19,35)(22,32,41,33)(25,39,42,43)(29,40,31,36);(1,2)(3,4)(6,9)(10,12)(13,15,20,27)(14,21,24,38)(16,18,26,19)(17,31,34,29)(22,39,41,43)(23,37,35,28)(25,32,42,33)(30,36,44,40);(5,7,8,11)(6,10,9,12)(13,16,18,27,20,26,19,15)(14,21,35,37,24,38,23,28)(17,31,32,25,34,29,33,42)(22,40,44,39,41,36,30,43);(2,4)(13,17,19,33,20,34,18,32)(14,22,23,30,24,41,35,44)(15,25,26,31,27,42,16,29)(21,36,28,39,38,40,37,43);(6,9)(10,12)(13,18,20,19)(14,23,24,35)(15,26,27,16)(17,32,34,33)(21,37,38,28)(22,30,41,44)(25,31,42,29)(36,43,40,39)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fgi,d^2=g,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7468416);