Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394628 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14718 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60367 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2170 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)(4,9)(5,7)(6,11)(10,17)(12,18)(13,27)(14,20)(15,19)(16,24)(21,28)(22,26)(23,29)(25,31)(30,32);(2,7)(3,12)(5,17)(8,20)(9,19)(10,11)(13,25)(14,26)(15,29)(21,27)(23,24)(28,32);(1,3,5,12,6,14,17,26)(2,8,10,22,11,20,7,18)(4,13,15,25,16,28,29,32)(9,21,23,31,24,27,19,30);(1,4)(2,9)(3,13)(5,15)(6,16)(7,19)(8,21)(10,23)(11,24)(12,25)(14,28)(17,29)(18,30)(20,27)(22,31)(26,32)Polycyclic presentation:
2^9|c^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,g^b=gi,f^c=fi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72306351, #72306445, #72306472, … |
| maximal normal subgroup of | #72306351, #72306445, #72306472, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301824);