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Cayley diagram

C4×#560 #34351

order256
nilpotency class2
isoclinismisoclinic class of #560
automorphismAut(G) order 196608
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C28

Aliases

direct productC4×#560

Subgroups

subgroupG/H
centerC22×C4C24
commutatorC22C24×C4
Frattini, socle, cosocleC23C25
prime radicalC2×#560C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C23 ▷ C1

1284 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 5 generators on 20 points:

(1,2,3,4);(5,6)(7,11)(8,10)(9,12)(13,14)(15,17)(16,18)(19,20);(5,7)(6,9)(8,12)(10,11)(13,15)(14,17)(16,19)(18,20);(7,12)(9,11)(14,18)(15,19);(6,10)(7,12)(14,18)(17,20)

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofC4×#560⋊C3, C4×#4626⋊C2
maximal subgroup of#129610, C4×#2200, #511912, …
maximal normal subgroup of#129610, C4×#2200, #511912, …
maximal quotient ofC4×#2375, C4×#2377, C4×#2376, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 27339);