Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500797 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24260 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| prime radical | #33553 | C2 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1041 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,8)(3,12)(4,16)(5,10)(9,22)(11,21)(13,29)(14,31)(15,28)(17,32)(18,25)(20,24)(23,27)(26,30);(1,2,5,8,7,4,10,16)(3,9,14,20,15,13,23,30)(6,17,18,32,19,11,25,21)(12,24,27,22,28,26,31,29);(1,3)(2,9)(4,13)(5,14)(6,12)(7,15)(8,20)(10,23)(11,24)(16,30)(17,26)(18,27)(19,28)(21,22)(25,31)(29,32);(1,4,5,16,7,2,10,8)(3,13,14,30,15,9,23,20)(6,17,18,32,19,11,25,21)(12,26,27,29,28,24,31,22)Polycyclic presentation:
2^9|b^2=e,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=ghi,c^b=cg,f^b=fh,h^c=hi,g^f=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72236883, #72236885, #72236890, … |
| maximal normal subgroup of | #72236890, #72237523, #72237525, … |
| maximal quotient of | #72312398, #72312487, #72312593, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407993);