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Cayley diagram

C2×#384 #2284

order128
nilpotency class2
isoclinismisoclinic class of #120
automorphismAut(G) order 786432
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4

Aliases

direct productC2×#384

Subgroups

subgroupG/H
center, socle, prime radicalC24C23
commutatorC22C23×C4
Frattini, cosocleC23C24
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

444 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 4 generators on 14 points:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

Frattini subgroup of#94101, #94105, #94196, …
Fitting subgroup ofC2×#384⋊C3, #11711327, #11711335, …
maximal subgroup ofC2×#1296, #8040, #8043, …
maximal normal subgroup ofC2×#1296, #8040, #8043, …
maximal quotient ofC2×C4⋊#95, C4×#384, C2×#1444, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 1016);