Finite Group Databasebeta

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Cayley diagram

C2×#431 #2917

order128
nilpotency class2
isoclinismisoclinic class of D4
automorphismAut(G) order 16384
homologyAbelianization (H1) = C23×C8 · Schur multiplier (H2) = C24×C4

Aliases

direct productC2×#431

Subgroups

subgroupG/H
centerC2×C42C22
commutatorC2C23×C8
Frattini, cosocleC2×C4C24
socleC23C22×C4
prime radicalC24C2×C4
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C2×C42 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

264 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 4 generators on 18 points:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

Frattini subgroup of#92964, #92966, #92968, …
maximal subgroup of#7848, #7849, #7850, …
maximal normal subgroup of#7848, #7849, #7850, …
maximal quotient ofC2×#1744, C2×C4×C22⋊C8, C2×C4×C4⋊C8, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 1649);