Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682500 |
| 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×#3180 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1894 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6)(7,13)(8,12)(9,11)(10,15)(14,20)(16,18)(17,19)(21,22,26,29)(23,30,32,36)(24,28,25,34)(27,31,35,33);(1,2)(3,4)(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18);(2,4)(21,23,25,33,26,32,24,31)(22,27,28,30,29,35,34,36);(6,11)(10,13)(12,19)(18,20)(21,24,26,25)(22,28,29,34)(23,31,32,33)(27,30,35,36);(2,4)(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(23,32)(27,35)(30,36)(31,33)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^c=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72308036, #72308060, #72308114, … |
| maximal normal subgroup of | #72308036, #72308060, #72308114, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6593081);