Finite Group Databasebeta

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Cayley diagram

C2×#8008 #181320

order512
nilpotency class3
isoclinismisoclinic class of #1284
automorphism?
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C27

Aliases

direct productC2×#8008

Subgroups

subgroupG/H
centerC22×C4#95
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
socleC23C2×#95
prime radicalC26Q8
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C25×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

2986 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 4 generators on 26 points:

(1,2);(3,4)(5,9,10,7)(6,8)(11,12,14,18,16,20,15,19)(13,21,22,17,24,25,23,26);(3,5)(4,7)(6,10)(8,9)(11,13,15,23,16,24,14,22)(12,17,18,25,20,26,19,21);(11,14,16,15)(12,18,20,19)(13,22,24,23)(17,25,26,21)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#201929, C2×#181326, C2×#188834, …
maximal normal subgroup ofC2×#201929, C2×#181326, C2×#188834, …
maximal quotient of#72309190

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88516);