Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395822 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14727 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1830 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)(3,13,16,24)(4,9)(5,10)(6,11)(7,12)(8,15,23,20)(14,25,30,29)(17,22,31,32)(18,26)(19,28)(21,27);(1,3)(2,8)(4,17)(5,20)(6,15)(7,16)(9,25)(10,24)(11,13)(12,23)(14,18)(19,31)(21,30)(22,27)(26,32)(28,29);(1,4,5,18,7,19,6,21)(2,9,10,26,12,28,11,27)(3,14,15,17,16,30,20,31)(8,22,13,29,23,32,24,25);(1,3)(2,8)(4,14)(5,15)(6,20)(7,16)(9,22)(10,13)(11,24)(12,23)(17,18)(19,30)(21,31)(25,27)(26,29)(28,32)Polycyclic presentation:
2^9|a^2=f,c^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^c=hi,f^d=fi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72216029, #72216154, #72216203, … |
| maximal normal subgroup of | #72216029, #72216154, #72216203, … |
| maximal quotient of | C2×#395822, #72255641, #72255812, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 303018);