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Cayley diagram

C4×#401 #13866

order256
nilpotency class2
isoclinismisoclinic class of #392
automorphismAut(G) order 1376256
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C25

Aliases

direct productC4×#401

Subgroups

subgroupG/H
centerC23×C4C23
commutatorC23C23×C4
Frattini, socle, prime radical, cosocleC24C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C24 ▷ C1

611 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 28
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofC4×#401⋊C3, C4×#401⋊C7
maximal subgroup ofC8×#401, #359421, #379688, …
maximal normal subgroup ofC8×#401, #359421, #379688, …
maximal quotient ofC4×#1443, #154481, #154613, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6854);