Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61569 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | C2×#3481 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3218 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,11)(4,9)(5,15)(6,10)(7,12)(8,13)(14,16)(17,18,21,26,22,24,20,25)(19,27,29,23,30,32,28,31);(1,3)(2,7)(4,12)(5,14)(6,13)(8,9)(10,16)(11,15)(17,19)(18,23)(20,28)(21,29)(22,30)(24,31)(25,27)(26,32);(1,3)(2,8)(4,14)(5,12)(6,13)(7,9)(10,11)(15,16)(18,24)(23,31)(25,26)(27,32);(1,4,6,5)(2,9,10,15)(3,12,13,14)(7,8,16,11)(17,20,22,21)(18,25,24,26)(19,28,30,29)(23,27,31,32);(2,10)(7,16)(8,11)(9,15)Polycyclic presentation:
2^9|a^2=g,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=ch,e^a=ehi,f^a=fh,f^b=fh,d^c=dhi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72280629, #72280709, #72339789, … |
| maximal normal subgroup of | #72280629, #72280709, #72339789, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250337);