Finite Group Databasebeta

← Prev (#2278) Next (#2280) →

Cayley diagram

C2×#381 #2279

order128
nilpotency class2
isoclinismisoclinic class of #120
automorphismAut(G) order 262144
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C27

Aliases

direct productC2×#381

Subgroups

subgroupG/H
center, socleC24C23
commutatorC22C23×C4
Frattini, cosocleC23C24
prime radicalC25C22
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

572 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);(4,6)(7,8,9,10)(12,15)(13,17);(11,12)(13,16)(14,15)(17,18);(3,4)(5,6)(7,8)(9,10)(11,13,14,17)(12,16,15,18)

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

derived subgroup of#72237977, #72237979, #72237981, …
Frattini subgroup of#94070, #94074, #94292, …
maximal subgroup of#7913, C2×#1284, #7943, …
maximal normal subgroup of#7913, C2×#1284, #7943, …
maximal quotient ofC2×C22⋊#95, C2×#1434, C4×#381, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 1011);