Finite Group Databasebeta

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Cayley diagram

C2×#14426 #520749

order512
nilpotency class2
isoclinismisoclinic class of #2668
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C42

Aliases

direct productC2×#14426

Subgroups

subgroupG/H
center, socle, prime radicalC25C24
commutatorC23C24×C4
Frattini, cosocleC24C25
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C24 ▷ C1

3594 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 512

A minimal faithful permutation representation — 5 generators on 26 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#2807614, C2×#579776, C2×#1084930
maximal normal subgroup ofC2×#2807614, C2×#579776, C2×#1084930
maximal quotient ofC2×#154771

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 427945);