Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453995 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2812 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16797 |
| prime radical | #23412 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1093 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)(3,15)(4,16)(5,11)(6,13)(7,12)(8,14)(9,17)(10,21)(18,19)(20,29)(22,28)(23,32)(24,31)(25,27)(26,30);(1,3,6,25,8,18,7,17)(2,9,13,15,14,27,12,19)(4,11,21,30,22,31,20,32)(5,16,26,10,24,28,23,29);(1,4)(2,10)(3,11)(5,9)(6,21)(7,20)(8,22)(12,16)(13,28)(14,29)(15,26)(17,32)(18,31)(19,23)(24,27)(25,30);(1,5)(2,11)(3,16)(4,19)(6,26)(7,23)(8,24)(9,21)(10,25)(12,32)(13,30)(14,31)(15,22)(17,29)(18,28)(20,27)Polycyclic presentation:
2^9|b^2=egh,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=hi,e^d=ei,g^d=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72339026, #72339032, #72339041 |
| maximal normal subgroup of | #72339026, #72339032, #72339041 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 378656);