Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2796 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,25)(16,22)(17,34)(18,24)(19,23)(20,29)(21,33)(26,36)(27,41)(28,38)(30,42)(31,37)(32,39)(35,40)(43,44);(5,7)(6,9)(8,12)(10,11)(13,15,18,29)(14,20,24,25)(16,31,33,27)(17,30,19,28)(21,37,22,41)(23,42,34,38)(26,35,44,32)(36,39,43,40);(2,4)(13,16)(14,21)(15,26)(17,32)(18,33)(19,35)(20,36)(22,24)(23,39)(25,43)(27,30)(28,31)(29,44)(34,40)(37,42)(38,41);(2,4)(6,10)(7,12)(13,16)(14,22)(15,27)(17,35)(18,33)(19,32)(20,37)(21,24)(23,39)(25,41)(26,30)(28,44)(29,31)(34,40)(36,42)(38,43);(13,17,18,19)(14,23,24,34)(15,28,29,30)(16,32,33,35)(20,38,25,42)(21,39,22,40)(26,31,44,27)(36,41,43,37)Polycyclic presentation:
2^9|a^2=gi,b^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410360);