Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #106766 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22⋊#95 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1065 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,11,24,3,15,17)(4,19,7,28,26,31,9,30)(5,13,22,32,8,14,27,25)(10,23,18,21,16,29,12,20);(1,3,7,18,24,2,9,12)(4,19,6,25,26,31,15,32)(5,14,23,28,8,13,29,30)(10,22,17,21,16,27,11,20);(4,20)(6,22)(10,19)(11,32)(15,27)(16,31)(17,25)(21,26)Polycyclic presentation:
2^9|a^2=f,b^2=g,f^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,d^b=di,e^b=eh,f^b=fh,d^c=dh,f^c=fh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72210458, #72213613, #72213681, … |
| maximal normal subgroup of | #72210458, #72213613, #72213681, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 13989);