Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624812 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ C8×#41 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1432 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,10)(4,8)(5,14)(6,9)(7,12)(11,16)(13,15)(18,22)(19,26)(20,27)(23,29)(24,28)(25,32)(30,31);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,10)(14,16)(18,23)(22,29)(25,30)(31,32);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,18,24,32,20,25,19,22,21,23,26,31,27,30,28,29);(3,12)(7,10)(11,13)(15,16);(1,5,6,4)(2,8,9,14)(3,13,12,11)(7,15,10,16)Polycyclic presentation:
2^9|c^2=fghi,e^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,c^b=ci,d^b=df80 conjugacy classes — view the full character table.
| maximal subgroup of | #72319253, #72320020, #72320286, … |
| maximal normal subgroup of | #72319253, #72320020, #72320286, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532294);