Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #461177 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16382 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
927 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)(4,17,18,8)(5,11)(6,10)(7,15)(9,19,25,20)(13,27,14,22)(16,29)(21,26)(23,30,28,31)(24,32);(1,3)(2,7,10,12)(4,14)(5,16)(6,15)(8,22,17,27)(9,23,25,28)(11,24,26,29)(13,18)(19,31)(20,30)(21,32);(1,4)(2,8)(3,13,15,14)(5,19)(6,18)(7,22,12,27)(9,11)(10,17)(16,30,32,31)(20,21)(23,29,28,24)(25,26);(1,4)(2,9)(3,14)(5,20)(6,18)(7,23)(8,11)(10,25)(12,28)(13,15)(16,30)(17,26)(19,21)(22,24)(27,29)(31,32)Polycyclic presentation:
2^9|a^2=efg,b^2=ef,c^2=e,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,d^c=de,g^c=gi,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72286134, #72286154, #72286246, … |
| maximal normal subgroup of | #72286134, #72286154, #72286246, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 368373);