Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24472 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14023 |
| prime radical | #60415 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1689 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,12,5,9)(3,13,14,8,17,23,16,24)(4,18,19,10,21,25,20,26)(15,22,28,31,30,32,29,27);(1,3,6,14,7,17,5,16)(2,8,11,23,12,24,9,13)(4,15,19,28,21,30,20,29)(10,22,25,31,26,32,18,27);(2,9,12,11)(3,14,17,16)(10,18,26,25)(15,28,30,29);(1,4)(2,10)(3,15)(5,20)(6,19)(7,21)(8,22)(9,18)(11,25)(12,26)(13,27)(14,28)(16,29)(17,30)(23,31)(24,32)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=fg,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^b=gi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72279719, #72286681, #72287295, … |
| maximal normal subgroup of | #72279719, #72286681, #72287295, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 276076);