Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394285 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14716 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2608 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)(3,6)(4,9)(5,10)(7,11)(8,12)(13,17)(14,19)(15,18)(16,28)(20,26)(21,25)(22,24)(23,30)(27,31)(29,32);(1,3,7,12,8,11,6,2)(4,15,22,24,18,9,17,13)(5,14,21,26,20,25,19,10)(16,28,31,32,30,23,29,27);(1,4)(2,9)(3,13)(5,16)(6,17)(7,22)(8,18)(10,23)(11,24)(12,15)(14,27)(19,29)(20,30)(21,31)(25,32)(26,28);(1,5)(2,10)(3,14)(4,16)(6,19)(7,21)(8,20)(9,23)(11,25)(12,26)(13,27)(15,28)(17,29)(18,30)(22,31)(24,32)Polycyclic presentation:
2^9|b^2=ei,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,g^c=gi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72237692, #72240440, #72240453, … |
| maximal normal subgroup of | #72237692, #72240440, #72240453, … |
| maximal quotient of | #72237696, C2×#394286, #72312690, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301482);