Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2202 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C2×Q8 | C2×C42 |
| Frattini, prime radical, cosocle | C22×#41 | C23 |
| socle | C22 | #1889 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
697 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 697 conjugacy classes of subgroups — too many to draw. Only its 154 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,4,3)(5,6,11,17)(7,19,22,34)(8,15,9,25)(10,18,12,16)(13,23,30,20)(14,26,33,24)(21,32,27,28)(29,35,31,36);(5,7,12,21,11,22,10,27)(6,13,18,31,17,30,16,29)(8,23,24,35,9,20,26,36)(14,32,15,19,33,28,25,34);(1,3,4,2)(5,8,11,9)(6,14,17,33)(7,20,22,23)(10,26,12,24)(13,28,30,32)(15,16,25,18)(19,31,34,29)(21,36,27,35)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,d^2=i,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,g^a=gi,h^a=hi,c^b=ci,e^b=ehi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72312047, #72312068, #72312086, … |
| maximal normal subgroup of | #72312047, #72312068, #72312086, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54542);