Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1776 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18,21,19,20,24,22,23);(2,4)(5,6)(7,8)(17,19,21,24,20,23,22,18);(6,8)(10,13)(14,16)(18,23)(19,24)(21,22);(2,4)(9,11,12,15)(10,14,13,16)(18,24)(19,23)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=fg,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,f^c=fi,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72305517, #72305527, #72305543, … |
| maximal normal subgroup of | #72305517, #72305527, #72305543, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 279469);