Finite Group Databasebeta

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Cayley diagram

C2×#14501 #520875

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

Aliases

direct productC2×#14501

Subgroups

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

4066 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);(4,5)(7,8,9,10)(11,12,15,13)(14,17,18,16);(12,13)(14,18)(19,20,22,24)(21,23,25,26);(3,4,6,5)(11,13,15,12)(14,17,18,16)(19,20)(21,23)(22,24)(25,26);(3,5,6,4)(11,14,15,18)(12,16,13,17)(19,21,22,25)(20,23,24,26)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#148172, #72266899, #72266910, …
maximal normal subgroup ofC2×#148172, #72266899, #72266910, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 428071);