Finite Group Databasebeta

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Cayley diagram

C23×#108 #60110

order256
nilpotency class3
isoclinismisoclinic class of D8
automorphismAut(G) order 5505024
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C29

Aliases

direct productC23×#108

Subgroups

subgroupG/H
centerC23×C4D4
commutatorC4C24×C4
Frattini, cosocleC2×C4C25
socleC24C4⋊C4
prime radicalC25Q8
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C4 ◁ C24×C4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

1423 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 5 generators on 22 points:

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

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofC2×A4×#108, #108×AGL(1,𝔽8)
maximal subgroup of#123877, #123878, #123881, …
maximal normal subgroup of#123877, #123878, #123881, …
maximal quotient ofC23×#334, C23×#335, C22×#1564, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 53098);