Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #472681 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16831 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
812 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)(3,12,14,7)(4,8)(5,11)(6,9)(10,20)(13,21,29,28)(15,23,31,22)(16,32,30,24)(17,27)(18,25)(19,26);(1,3)(2,7,11,12)(4,16,17,30)(5,14)(6,15)(8,24)(9,22,10,23)(13,18,29,19)(20,31)(21,25)(26,28)(27,32);(1,4)(2,8)(3,13,14,29)(5,17)(6,18)(7,21,12,28)(9,25)(10,26)(11,27)(15,16,31,30)(19,20)(22,32,23,24);(2,9)(4,17)(6,20)(7,22)(8,25)(10,11)(12,23)(15,31)(16,30)(21,24)(26,27)(28,32)Polycyclic presentation:
2^9|a^2=f,b^2=efgh,c^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,d^c=de,h^c=hi,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72237071, #72237078, #72237086, … |
| maximal normal subgroup of | #72237071, #72237078, #72237086, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 379877);