Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139980 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1889 |
| commutator | C2×C4⋊C4 | C22×C4 |
| Frattini, cosocle | C2×C4×Q8 | C23 |
| prime radical | #18186 | C2 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1 | |
492 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 492 conjugacy classes of subgroups — too many to draw. Only its 66 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,6)(3,10)(4,7)(9,14)(11,12)(13,15)(17,18,19,23,22,27,20,25)(21,30,29,32,31,26,28,24);(1,2,7,14,5,9,4,6)(3,8,10,15,12,16,11,13)(18,23)(19,20)(24,26)(25,27)(28,29)(30,32);(1,3)(2,8)(4,11)(5,12)(6,13)(7,10)(9,16)(14,15)(17,19)(18,24)(20,22)(21,29)(23,30)(25,26)(27,32)(28,31)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=gi,e^2=gi,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eghi,h^a=hi,c^b=ci,d^b=di,e^b=ehi,f^b=fi,d^c=dh,e^c=egi,f^c=fh,e^d=ei,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72308360, #72308382, #72308394 |
| maximal normal subgroup of | #72308360, #72308382, #72308394 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47212);