Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #466062 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23400 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1065 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)(3,4)(5,21)(6,13)(7,12)(8,14)(9,10)(11,25)(15,18)(16,19)(17,20)(22,26)(23,27)(24,28)(29,32)(30,31);(1,3)(2,9,14,19)(4,13,20,12)(5,15)(6,10)(7,16)(8,17)(11,22,21,32)(18,27,31,28)(23,29)(24,26)(25,30);(1,4,6,19,8,20,7,9)(2,10,12,17,14,16,13,3)(5,22,23,31,25,32,24,18)(11,15,27,29,21,30,28,26);(1,5,7,23,8,25,6,24)(2,11,13,27,14,21,12,28)(3,15,16,29,17,30,10,26)(4,18,9,22,20,31,19,32)Polycyclic presentation:
2^9|b^2=fg,c^2=e,d^2=f,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,h^b=hi,d^c=de,g^c=gi,h^c=hi,e^d=ei,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72237063, #72237064, #72237082, … |
| maximal normal subgroup of | #72237063, #72237064, #72237082, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 373258);