Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470137 |
| 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 | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2056 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,3)(5,6)(7,9)(8,10)(11,12)(13,14,19,23)(15,20,26,36)(16,27,29,37)(17,22,18,31)(21,32,38,28)(24,39,42,44)(25,34,30,35)(33,40,43,41);(1,2,4,3)(5,7)(6,9)(8,11)(10,12)(13,15)(14,20)(16,24)(17,30)(18,25)(19,26)(21,33)(22,35)(23,36)(27,39)(28,41)(29,42)(31,34)(32,40)(37,44)(38,43);(1,3,4,2)(7,11)(9,12)(13,16,19,29)(14,21,23,38)(15,24,26,42)(17,28,18,32)(20,33,36,43)(22,37,31,27)(25,40,30,41)(34,39,35,44);(7,11)(9,12);(6,10)(9,12)(13,17,19,18)(14,22,23,31)(15,25,26,30)(16,28,29,32)(20,34,36,35)(21,37,38,27)(24,40,42,41)(33,39,43,44)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^b=ei,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6377927);