Finite Group Databasebeta

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Cayley diagram

C2×#14476 #520832

order512
nilpotency class2
isoclinismisoclinic class of #2668
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C25×C43

Aliases

direct productC2×#14476

Subgroups

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

3226 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 512

A minimal faithful permutation representation — 5 generators on 30 points:

(1,2);(3,4,5,6)(15,16,19,17)(18,21,22,20)(24,25)(28,29);(8,11)(12,14)(16,17)(18,22)(23,24,27,25)(26,29,30,28);(7,8)(9,12)(10,11)(13,14)(15,17,19,16)(18,21,22,20)(23,25,27,24)(26,29,30,28);(7,9,10,13)(8,12,11,14)(15,18,19,22)(16,20,17,21)(23,26)(24,28)(25,29)(27,30)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#3078577, C2×#581275, C2×#4315656
maximal normal subgroup ofC2×#3078577, C2×#581275, C2×#4315656
maximal quotient ofC2×#154771

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 428028);