Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61569 |
| automorphism | Aut(G) order 131072 |
| 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 | |
3220 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)(3,10)(4,6)(5,8)(7,11)(9,13)(12,15)(14,16)(17,18,21,25,22,24,20,29)(19,26,27,23,28,32,30,31);(2,6)(4,13)(7,14)(8,9)(10,15)(12,16)(17,19)(18,23)(20,30)(21,27)(22,28)(24,31)(25,32)(26,29);(1,3)(2,7)(4,12)(5,11)(6,14)(8,10)(9,15)(13,16)(18,24)(23,31)(25,29)(26,32);(3,11)(7,10)(12,16)(14,15);(2,8)(6,9)(7,10)(14,15)Polycyclic presentation:
2^9|a^2=g,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 | #72280709, #72302206, #72317645, … |
| maximal normal subgroup of | #72280709, #72302206, #72317645, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250335);