Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470901 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1994 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)(3,4)(5,6)(7,12)(8,10)(9,11)(13,14,18,22)(15,23,26,39)(16,20,28,35)(17,31,24,21)(19,32,34,27)(25,38,29,42)(30,44,40,37)(33,43,36,41);(2,4)(7,8)(10,12)(16,28)(20,35)(25,29)(30,40)(33,36)(37,44)(38,42)(41,43);(5,7,9,8)(6,10,11,12)(13,15,24,32,18,26,17,27)(14,19,31,39,22,34,21,23)(16,29,40,41,28,25,30,43)(20,36,44,38,35,33,37,42);(5,8,9,7)(6,10,11,12)(13,16,18,28)(14,20,22,35)(15,25,26,29)(17,30,24,40)(19,33,34,36)(21,37,31,44)(23,38,39,42)(27,41,32,43);(6,11)(10,12)(15,26)(16,28)(19,34)(20,35)(23,39)(27,32)(30,40)(37,44)Polycyclic presentation:
2^9|a^2=i,c^2=gi,d^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dhi,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6390375);