Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417690 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2669 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1753 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4)(5,6,10,14,11,16,9,15)(7,17,8,12,18,20,19,13);(5,7)(6,12)(8,10)(9,19)(11,18)(13,16)(14,20)(15,17);(2,4)(5,6)(7,17)(8,13)(9,14)(10,15)(11,16)(12,19)(18,20);(5,8,10,7,11,19,9,18)(6,13,15,17,16,12,14,20)Polycyclic presentation:
2^9|a^2=f,d^2=ei,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72286950, #72287003, #72287060 |
| maximal normal subgroup of | #72286950, #72287003, #72287060 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327120);