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Cayley diagram

C2×#2955 #60179

order256
nilpotency class3
isoclinismisoclinic class of AGL(1,ℤ/8)
automorphismAut(G) order 32768
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C29

Aliases

direct productC2×#2955

Subgroups

subgroupG/H
centerC2×C8C2×D4
commutatorC4C24×C4
Frattini, cosocleC2×C4C25
socleC22C2×C4×D4
prime radicalC2×#575C2
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C8 ◁ C23×C8 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

1333 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 34
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 5 generators on 34 points:

(1,2);(3,4,8,13,9,12,7,21)(5,14,16,10,17,25,22,24)(6,11,19,27,20,26,18,31)(15,28,29,23,30,34,32,33);(3,5)(4,10)(6,15)(7,22)(8,16)(9,17)(11,23)(12,24)(13,25)(14,21)(18,32)(19,29)(20,30)(26,33)(27,34)(28,31);(3,6,8,19,9,20,7,18)(4,11,13,27,12,26,21,31)(5,15,16,29,17,30,22,32)(10,23,25,34,24,33,14,28);(4,12)(10,24)(11,26)(13,21)(14,25)(23,33)(27,31)(28,34)

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#129510, #147536, #147537, …
maximal normal subgroup of#129510, #147536, #147537, …
maximal quotient ofC2×C8×AGL(1,ℤ/8), C2×C8×#137, C8×#575, …
automorphism group of#1422

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 53167);