Finite Group Databasebeta

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Cayley diagram

C2×#9909 #207220

order512
nilpotency class3
isoclinismisoclinic class of #7654
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4

Aliases

direct productC2×#9909

Subgroups

subgroupG/H
center, socleC24C2×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

1950 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 128

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

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#208390
maximal normal subgroup ofC2×#208390

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 114416);