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 | |
2376 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)(15,24)(16,27)(17,22)(18,23)(19,20)(21,25)(26,37)(28,43)(29,40)(30,34)(31,44)(32,39)(33,36)(35,41)(38,42);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15,18,19)(14,20,23,24)(16,25,29,33)(17,30,32,26)(21,27,36,40)(22,37,39,34)(28,42,31,41)(35,44,38,43);(5,7)(6,9)(8,12)(10,11)(13,16,18,29)(14,21,23,36)(15,25,19,33)(17,31,32,28)(20,27,24,40)(22,38,39,35)(26,42,30,41)(34,44,37,43);(1,3,4,2)(7,12)(9,11)(13,17)(14,22)(15,26)(16,28)(18,32)(19,30)(20,34)(21,35)(23,39)(24,37)(25,41)(27,43)(29,31)(33,42)(36,38)(40,44);(6,10)(9,11)(14,23)(17,32)(20,24)(21,36)(26,30)(27,40)(28,31)(41,42)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=i,d^2=fi,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, 6903769);