Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480263 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C4⋊C4 | C24 |
| prime radical | C22×#127 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1418 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,8,5,6,3,7)(9,10)(11,17,18,24)(12,20,22,15)(13,16)(14,19)(21,23);(9,11)(10,14)(12,21)(13,18)(15,24)(16,23)(17,20)(19,22);(2,6)(7,8);(1,3,5,4)(2,7,6,8)(11,18)(14,23)(17,24)(19,21)Polycyclic presentation:
2^9|a^2=f,d^2=h,e^2=i,f^2=h,b^a=be,c^a=ch,e^a=eg,g^a=gi,d^b=di,e^b=ei,f^b=fgi,f^e=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72230994, #72231054, #72231069, … |
| maximal normal subgroup of | #72230994, #72231054, #72231069, … |
| maximal quotient of | C2×#480400, #72261213, #72261807, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387596);