Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33688 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#2213 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
759 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(6,12)(7,13,17,20)(8,10)(9,14)(15,19,16,18);(1,2,3,4)(5,6,8,12,11,14,10,9)(7,13,16,19,17,20,15,18);(5,7)(6,13)(8,16)(9,18)(10,15)(11,17)(12,19)(14,20);(6,14)(7,15,17,16)(9,12)(13,19,20,18)Polycyclic presentation:
2^9|a^2=g,b^2=e,d^2=g,e^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,d^b=di,d^c=dh,f^c=fhi,g^c=gi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72213969, #72214039, #72214066, … |
| maximal normal subgroup of | #72213969, #72214039, #72214066, … |
| maximal quotient of | #72214602, #72216353, #72216361, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403046);