Finite Group Databasebeta

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Cayley diagram

C2×#8009 #181322

order512
nilpotency class3
isoclinismisoclinic class of #7943
automorphism?
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C27

Aliases

direct productC2×#8009

Subgroups

subgroupG/H
center, socleC23C2×#95
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC26Q8
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

2878 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)(5,9,10,7)(6,8)(11,12,14,19,16,20,15,18)(13,21,23,17,22,26,24,25);(3,5)(4,7)(6,10)(8,9)(11,13,15,24,16,22,14,23)(12,17,19,26,20,25,18,21);(5,10)(7,9)(13,22)(17,25)(21,26)(23,24)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#201940, C2×#201920, C2×#181326
maximal normal subgroup ofC2×#201940, C2×#201920, C2×#181326
maximal quotient of#72309129, #72309166

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88518);