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Cayley diagram

C2×#1397 #13547

order256
nilpotency class2
isoclinismisoclinic class of D4
automorphismAut(G) order 8192
homologyAbelianization (H1) = C2×C4×C16 · Schur multiplier (H2) = C23

Aliases

direct productC2×#1397

Subgroups

subgroupG/H
centerC22×C16C22
commutatorC2C2×C4×C16
Frattini, cosocleC2×C16C23
socle, prime radicalC23C2×C16
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C22×C16 ◁ G
LECSG ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1

127 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 38
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 3 generators on 38 points:

(1,2);(7,8,9,14,10,15,17,26,11,16,18,27,20,28,29,36,12,13,19,23,21,24,30,33,22,25,31,34,32,35,37,38);(3,4,5,6)(8,13)(14,23)(15,24)(16,25)(26,33)(27,34)(28,35)(36,38)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

Frattini subgroup of#72234375, #72234385, #72234403, …
maximal subgroup of#93990, #93993, #94023, …
maximal normal subgroup of#93990, #93993, #94023, …
maximal quotient ofC2×#7329, C2×#7331, #149506, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6535);