Finite Group DatabasebetaFinite Group Databasebeta| order | 27008 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C211 · Schur multiplier (H2) = C25 |
| direct product | #1794×C211 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C211 | #120 |
| commutator | C2×C4 | C22×C4×C211 |
| Frattini, cosocle | C22×C4 | C23×C211 |
| socle | C2×C211 | #379 |
| prime radical | #585×C211 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C211 ◁ C22×C4×C211 ◁ G | |
548 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 548 conjugacy classes of subgroups — too many to draw. Only its 132 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 243 |
| transitive degree | μt(G) = 6752 |
A minimal faithful permutation representation — 3 generators on 243 points:
(1,2,6,10,7,11,5,18)(3,12,14,8,15,23,19,22)(4,9,16,25,17,26,20,24)(13,28,31,32,30,27,29,21)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243);(1,3)(2,8)(4,13)(5,19)(6,14)(7,15)(9,21)(10,23)(11,22)(12,18)(16,31)(17,30)(20,29)(24,27)(25,28)(26,32);(1,4)(2,9)(3,13)(5,20)(6,16)(7,17)(8,21)(10,25)(11,26)(12,27)(14,31)(15,30)(18,24)(19,29)(22,32)(23,28)6752 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27008, 526);