Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 | #1999 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1776 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)(7,11)(10,12)(13,14)(15,24)(16,28)(17,33)(18,23)(19,27)(20,26)(21,32)(22,36)(25,39)(29,37)(30,40)(31,41)(34,38)(35,42)(43,44);(6,9)(7,11)(13,15,18,26)(14,19,23,38)(16,29,32,43)(17,34,36,27)(20,33,24,22)(21,39,28,40)(25,35,30,31)(37,42,44,41);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,30)(17,27)(18,26)(19,22)(21,40)(23,24)(25,32)(28,39)(29,35)(31,43)(33,38)(34,36)(37,41)(42,44);(6,9)(10,12)(13,16,18,32)(14,21,23,28)(15,25,26,30)(17,35,36,31)(19,37,38,44)(20,39,24,40)(22,42,33,41)(27,43,34,29);(5,7,8,11)(6,10,9,12)(15,26)(19,38)(20,24)(25,30)(27,34)(29,43)(37,44)(39,40)Polycyclic presentation:
2^9|a^2=gi,b^2=i,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^d=fi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7055694);