Finite Group Databasebeta

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Cayley diagram

C2×#11433 #480001

order512
nilpotency class4
isoclinismisoclinic class of #2127
automorphism?
homologyAbelianization (H1) = C23×C8 · Schur multiplier (H2) = C24×C42

Aliases

direct productC2×#11433

Subgroups

subgroupG/H
center, socleC23C2×#99
commutatorC2×C4C23×C8
Frattini, cosocleC2×C4⋊C4C24
prime radicalC24×D4C4
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

2238 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 4 generators on 26 points:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#480265, C2×#129232, #72292791
maximal normal subgroup ofC2×#480265, C2×#129232, #72292791
maximal quotient of#72292791, #72293071

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387197);