Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle | C23 | C23×C4 |
| prime radical | C22×D4 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
369 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 369 conjugacy classes of subgroups — too many to draw. Only its 151 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 — 4 generators on 32 points:
(1,2)(3,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18,21,25,22,19,23,27)(20,24,29,31,28,26,30,32);(1,3,6,2)(4,7,5,8)(17,19,21,27,22,18,23,25)(20,24,29,31,28,26,30,32);(1,4,6,5)(2,7,3,8)(10,13)(14,16)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32);(1,5,6,4)(2,7,3,8)(9,11,12,15)(10,14,13,16)(18,19)(20,28)(25,27)(29,30)Polycyclic presentation:
2^8|a^2=g,b^2=eg,c^2=e,d^2=ef,g^2=h,b^a=be,c^a=cf,d^a=deh,c^b=ceh,d^b=dh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#22396, #6376970, #6376972, … |
| maximal normal subgroup of | C2×#22396, #6376970, #6376972, … |
| maximal quotient of | #355318, #355672, #355708, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15384);