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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1920 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(7,11)(10,12)(13,14,20,24,19,25,18,35)(15,26,30,42,29,21,28,38)(16,31,32,39,17,23,34,22)(27,40,43,36,33,37,44,41);(1,2,3,4)(6,9)(7,11)(13,15)(14,21)(16,27)(17,33)(18,28)(19,29)(20,30)(22,36)(23,40)(24,38)(25,26)(31,37)(32,43)(34,44)(35,42)(39,41);(5,6)(7,10)(8,9)(11,12)(13,16,19,17)(14,22,25,39)(15,27,29,33)(18,32,20,34)(21,36,26,41)(23,35,31,24)(28,43,30,44)(37,38,40,42);(2,4)(6,9)(10,12)(13,17,19,16)(14,23,25,31)(15,27,29,33)(18,32,20,34)(21,37,26,40)(22,35,39,24)(28,44,30,43)(36,38,41,42);(5,7,8,11)(6,10,9,12)(13,18,19,20)(14,24,25,35)(15,28,29,30)(16,32,17,34)(21,38,26,42)(22,23,39,31)(27,43,33,44)(36,40,41,37)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=i,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7056362);