Finite Group Databasebeta

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Cayley diagram

C2×#45507 #7701964

order512
nilpotency class2
isoclinismisoclinic class of #45438
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC2×#45507

Subgroups

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

6812 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#7837053, C2×#7701979, C2×#7837038
maximal normal subgroup ofC2×#7837053, C2×#7701979, C2×#7837038

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7609160);