Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499849 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2002 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 2002 conjugacy classes of subgroups — too many to draw. Only its 152 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 — 4 generators on 36 points:
(5,6)(7,17)(8,13)(9,12)(10,26)(11,16)(14,20)(15,19)(18,30)(21,27)(22,34)(23,32)(24,29)(25,28)(31,36)(33,35);(1,2)(3,4)(5,7,11,20)(6,12,16,28)(8,18,23,35)(9,10,25,19)(13,29,32,27)(14,26,17,15)(21,36,24,22)(30,31,33,34);(2,4)(5,8,11,23)(6,13,16,32)(7,18,20,35)(9,24,25,21)(10,22,19,36)(12,27,28,29)(14,33,17,30)(15,34,26,31);(5,9,11,25)(6,14,16,17)(7,19,20,10)(8,21,23,24)(12,15,28,26)(13,30,32,33)(18,22,35,36)(27,31,29,34)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=di,e^b=ei,f^b=fh,d^c=di,e^c=eh,e^d=ei,f^d=fhi,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72317572, #72317591, #72317677, … |
| maximal normal subgroup of | #72317572, #72317591, #72317677, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407050);