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Cayley diagram

C23×#125 #62661

order256
nilpotency class2
isoclinismisoclinic class of #120
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C213

Aliases

direct productC23×#125

Subgroups

subgroupG/H
center, socle, prime radicalC25C23
commutator, Frattini, cosocleC22C26
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C22 ▷ C1

4207 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 6 generators on 22 points:

(3,4);(5,6);(1,2)(3,4)(5,6);(9,13)(11,14)(15,16,18,20)(17,19,21,22);(7,8,10,12)(9,11,13,14)(15,16)(17,19)(18,20)(21,22);(7,9)(8,11)(10,13)(12,14)(15,17,18,21)(16,19,20,22)

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofC2×A4×#125, #125×AGL(1,𝔽8)
maximal subgroup of#154374, #154442, #154789, …
maximal normal subgroup of#154374, #154442, #154789, …
maximal quotient ofC23×#382, C23×#387, C23×#389, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 55649);