Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129424 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C23×Q8 | C23 |
| prime radical | C2×#525 | C4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C24×Q8 ◁ G | |
| LECS | G ▷ C23×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1699 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 1699 conjugacy classes of subgroups — too many to draw. Only its 146 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:
(5,6,9,14,10,13,25,31)(7,15,18,33,17,29,23,35)(8,19,21,32,22,12,24,30)(11,26,28,20,27,34,36,16);(1,2)(3,4)(5,7)(6,11)(8,20)(9,23)(10,17)(12,29)(13,27)(14,28)(15,19)(16,22)(18,25)(21,26)(24,34)(30,35)(31,36)(32,33);(2,4)(11,27)(15,29)(28,36)(33,35)Polycyclic presentation:
2^9|a^2=g,d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,d^b=di,e^b=ei,g^b=ghi,g^d=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72227409, #72281001, #72286559, … |
| maximal normal subgroup of | #72227409, #72281001, #72286559, … |
| maximal quotient of | #72227408, #72227419, #72267888, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54767);