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Cayley diagram

C2×#14467 #520810

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

Aliases

direct productC2×#14467

Subgroups

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

4370 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)(20,21)(24,25);(12,13)(14,18)(19,20,23,21)(22,25,26,24);(3,4,6,5)(11,13,15,12)(14,17,18,16)(19,21,23,20)(22,25,26,24);(3,5,6,4)(11,14,15,18)(12,16,13,17)(19,22)(20,24)(21,25)(23,26)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#148173, #72266774, #72266782, …
maximal normal subgroup ofC2×#148173, #72266774, #72266782, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 428006);