Finite Group Databasebeta

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Cayley diagram

C2×#39982 #7638020

order512
nilpotency class2
isoclinismisoclinic class of #37644
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC2×#39982

Subgroups

subgroupG/H
center, socleC24C25
commutator, Frattini, cosocleC23C26
prime radicalC25C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

5458 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 6 generators on 26 points:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#7837039, C2×#7837032, C2×#7638041
maximal normal subgroup ofC2×#7837039, C2×#7837032, C2×#7638041

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7545216);