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 | |
2318 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,14)(15,27)(16,32)(17,23)(18,26)(19,25)(20,24)(21,30)(22,33)(28,41)(29,38)(31,39)(34,37)(35,40)(36,44)(42,43);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15)(14,21)(16,28)(17,34)(18,29)(19,30)(20,31)(22,36)(23,41)(24,38)(25,27)(26,39)(32,37)(33,43)(35,44)(40,42);(5,7)(6,9)(8,12)(10,11)(13,16)(14,22)(15,28)(17,19)(18,33)(20,35)(21,36)(23,26)(24,32)(25,40)(27,42)(29,43)(30,34)(31,44)(37,38)(39,41);(1,3,4,2)(7,12)(9,11)(13,17,19,16)(14,23,25,32)(15,28,30,34)(18,33,20,35)(21,37,27,41)(22,24,40,26)(29,44,31,43)(36,39,42,38);(6,10)(9,11)(13,18,19,20)(14,24,25,26)(15,29,30,31)(16,33,17,35)(21,38,27,39)(22,32,40,23)(28,43,34,44)(36,37,42,41)Polycyclic presentation:
2^9|a^2=fi,d^2=f,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, 6903756);