Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420424 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2699 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15984 |
| prime radical | #60469 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1259 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,8,3,7,11)(4,13,23,15,18,9,17,25)(5,14,21,19,22,10,20,26)(16,24,29,28,30,27,31,32);(1,3)(2,8)(4,15)(5,19)(6,11)(7,12)(9,23)(10,21)(13,17)(14,20)(16,24)(18,25)(22,26)(27,30)(28,29)(31,32);(1,4)(2,9)(3,13)(5,16)(6,17)(7,23)(8,18)(10,24)(11,25)(12,15)(14,27)(19,28)(20,29)(21,31)(22,30)(26,32);(1,5,6,20,8,22,7,21)(2,10,11,26,3,14,12,19)(4,16,17,29,18,30,23,31)(9,24,25,32,13,27,15,28)Polycyclic presentation:
2^9|a^2=gh,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72319976, #72320000, #72320075, … |
| maximal normal subgroup of | #72319976, #72320000, #72320075, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 358252);