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 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1076 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7)(3,12)(4,8)(9,23)(10,19)(11,20)(13,28)(14,27)(15,22)(16,29)(18,24)(21,26)(25,32)(30,31);(1,2,8,20,6,11,4,7)(3,9,21,31,14,25,13,18)(5,16,22,19,17,10,15,29)(12,24,28,32,27,30,26,23);(1,3)(2,9)(4,13)(5,12)(6,14)(7,18)(8,21)(10,24)(11,25)(15,26)(16,30)(17,27)(19,23)(20,31)(22,28)(29,32);(1,2,8,20,6,11,4,7)(3,9,21,31,14,25,13,18)(5,10,22,29,17,16,15,19)(12,24,28,32,27,30,26,23)Polycyclic presentation:
2^9|b^2=ei,d^2=ei,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 | #72236868, #72236878, #72236890, … |
| maximal normal subgroup of | #72236890, #72237521, #72237526, … |
| maximal quotient of | #72312398, #72312542, #72312593, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407997);