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 | #45134 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2054 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,14)(15,26)(16,22)(17,25)(18,24)(19,23)(20,30)(21,32)(27,36)(28,40)(29,37)(31,38)(33,41)(34,39)(35,44)(42,43);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15)(14,20)(16,27)(17,29)(18,30)(19,31)(21,35)(22,40)(23,37)(24,26)(25,38)(28,33)(32,43)(34,44)(36,41)(39,42);(5,7,8,12)(6,9,10,11)(13,16,18,33)(14,21,24,39)(15,27,30,28)(17,32,19,34)(20,35,26,42)(22,25,41,23)(29,43,31,44)(36,37,40,38);(1,3,4,2)(7,12)(9,11)(13,16)(14,22)(15,28)(17,34)(18,33)(19,32)(20,36)(21,23)(24,41)(25,39)(26,40)(27,30)(29,43)(31,44)(35,38)(37,42);(6,10)(9,11)(13,17,18,19)(14,23,24,25)(15,29,30,31)(16,32,33,34)(20,37,26,38)(21,22,39,41)(27,43,28,44)(35,40,42,36)Polycyclic presentation:
2^9|a^2=fi,c^2=hi,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ehi,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,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, 6963282);