Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418530 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2676 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15453 |
| prime radical | #23412 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1354 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)(3,8,15,23)(4,17)(5,12)(6,11)(7,10)(9,19)(13,24,16,21)(14,29,30,32)(18,27)(20,26)(22,31,25,28);(1,3)(2,8,10,23)(4,14,18,30)(5,13)(6,16)(7,15)(9,25)(11,21,12,24)(17,32)(19,28,20,31)(22,26)(27,29);(3,13,15,16)(8,21,23,24)(14,28,30,31)(22,29,25,32);(1,4)(2,9)(3,14,15,30)(5,19)(6,20)(7,18)(8,22,23,25)(10,26)(11,17)(12,27)(13,28,16,31)(21,29,24,32)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=f,d^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,d^b=de,e^b=ei,f^b=fi,g^b=gi,e^d=ei,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72279915, #72286957, #72287267, … |
| maximal normal subgroup of | #72279915, #72286957, #72287267, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 339447);