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Cayley diagram

C2×#1844 #20794

order256
nilpotency class3
isoclinismisoclinic class of #465
automorphismAut(G) order 2097152
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4

Aliases

direct productC2×#1844

Subgroups

subgroupG/H
center, socle, prime radicalC24C2×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

679 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 256

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

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

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

Frattini subgroup of#72279908, #72279942
maximal subgroup of#109002, #109005, #109007, …
maximal normal subgroup of#109002, #109005, #109007, …
maximal quotient ofC2×#8805, C2×#8809, #191383, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 13782);