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 | C2×#3002 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2859 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,9)(8,10)(11,12)(13,14,19,23)(15,20,26,35)(16,28,30,36)(17,22,18,31)(21,32,37,29)(24,38,40,43)(25,34,27,41)(33,42,44,39);(1,2,3,4)(5,7)(6,9)(8,11)(10,12)(13,15)(14,20)(16,24)(17,25)(18,27)(19,26)(21,33)(22,34)(23,35)(28,38)(29,39)(30,40)(31,41)(32,42)(36,43)(37,44);(7,11)(9,12)(13,16,19,30)(14,21,23,37)(15,24,26,40)(17,29,18,32)(20,33,35,44)(22,36,31,28)(25,39,27,42)(34,43,41,38);(7,11)(9,12)(13,17,19,18)(14,22,23,31)(15,25,26,27)(16,29,30,32)(20,34,35,41)(21,36,37,28)(24,39,40,42)(33,43,44,38);(6,10)(9,12)(15,26)(20,35)(24,40)(25,27)(33,44)(34,41)(38,43)(39,42)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,d^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, 6377485);