Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471438 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16797 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23402 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1133 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)(5,10)(6,21)(7,11)(8,15)(9,20)(13,24)(14,29)(16,22)(18,26)(19,25)(23,30)(27,28)(31,32);(1,3)(2,8,11,12)(4,14)(5,13)(6,16)(7,15)(9,23,25,27)(10,22,21,24)(17,32,26,29)(18,31)(19,30)(20,28);(3,13)(5,6)(8,22)(10,21)(12,24)(14,30)(15,16)(17,26)(19,20)(23,32)(27,29)(28,31);(1,4)(2,9)(3,14)(5,20)(6,19)(7,18)(8,23)(10,26)(11,25)(12,27)(13,28)(15,31)(16,30)(17,21)(22,29)(24,32)Polycyclic presentation:
2^9|b^2=egh,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,e^b=ei,g^b=gi,d^c=de,g^c=gi,h^c=hi,f^d=fi,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72237068, #72237071, #72237092, … |
| maximal normal subgroup of | #72237068, #72237071, #72237092, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 378634);