Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667980 |
| 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 | C22×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1938 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22)(23,28)(24,33)(25,30)(26,35)(27,31)(29,34)(32,36);(1,2)(3,4)(5,7,9,15)(6,10,12,13)(8,14,16,20)(11,17,18,19)(21,23,27,24)(22,28,31,33)(25,34,26,32)(29,35,36,30);(2,4)(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(21,24,27,23)(22,29,31,36)(25,34,26,32)(28,35,33,30);(6,12)(10,13)(11,18)(17,19)(21,25,27,26)(22,30,31,35)(23,32,24,34)(28,36,33,29);(2,4)(7,15)(10,13)(14,20)(17,19)Polycyclic presentation:
2^9|b^2=fi,c^2=fi,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72307686, #72307862, #72308006, … |
| maximal normal subgroup of | #72307686, #72307862, #72308006, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6577134);