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Cayley diagram

C23×#106 #60101

order256
nilpotency class3
isoclinismisoclinic class of D8
automorphismAut(G) order 88080384
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C210

Aliases

direct productC23×#106

Subgroups

subgroupG/H
center, socle, prime radicalC25D4
commutatorC4C24×C4
Frattini, cosocleC2×C4C25
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C25 ◁ C24×C4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

1807 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 5 generators on 18 points:

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

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofC2×A4×#106, #106×AGL(1,𝔽8)
maximal subgroup of#123875, #123883, #146510, …
maximal normal subgroup of#123875, #123883, #146510, …
maximal quotient ofC23×#334, C22×#1566, C22×#1570, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 53089);