Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #427841 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2778 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15492 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
781 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)(3,12,14,7)(4,8,17,26)(5,9)(6,10)(11,20)(13,24)(15,23,19,22)(16,21)(18,25,30,27)(28,32)(29,31);(1,3)(2,7)(4,16,17,13)(5,19)(6,14)(8,24,26,21)(9,23)(10,12)(11,22)(15,20)(18,28,30,29)(25,31,27,32);(1,4)(2,8,10,26)(3,13)(5,18)(6,17)(7,21,12,24)(9,25,11,27)(14,16)(15,28)(19,29)(20,30)(22,31,23,32);(2,9,10,11)(7,22,12,23)(8,25,26,27)(21,31,24,32)Polycyclic presentation:
2^9|a^2=efg,b^2=ef,c^2=e,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,g^b=gi,h^b=hi,f^c=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72285398, #72285951, #72286126, … |
| maximal normal subgroup of | #72285398, #72285951, #72286126, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 340968);