Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476143 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2819 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #17098 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
870 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,14,8,12,7,13)(3,15,16,28,18,9,17,29)(4,19,21,10,23,32,22,27)(5,24,20,31,25,30,26,11);(1,3)(2,9)(4,20)(5,21)(6,17)(7,16)(8,18)(10,30)(11,19)(12,15)(13,29)(14,28)(22,25)(23,26)(24,27)(31,32);(1,4)(2,10,12,27)(3,5)(6,21)(7,22)(8,23)(9,11,15,31)(13,19,14,32)(16,20)(17,26)(18,25)(24,28,30,29);(1,5)(2,11,12,31)(3,4)(6,26)(7,20)(8,25)(9,27,15,10)(13,24,14,30)(16,22)(17,21)(18,23)(19,29,32,28)Polycyclic presentation:
2^9|a^2=ef,c^2=e,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=def,f^b=fi,g^b=gi,d^c=de,g^c=gi,f^d=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72319935, #72320000, #72320075, … |
| maximal normal subgroup of | #72319935, #72320000, #72320075, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 384044);