Finite Group Databasebeta

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Cayley diagram

C2×#9513 #201807

order512
nilpotency class3
isoclinismisoclinic class of #9495
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C27

Aliases

direct productC2×#9513

Subgroups

subgroupG/H
center, socleC23#386
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×#41C22
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

2202 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#213912, #72238540, #72238542, …
maximal normal subgroup ofC2×#213912, #72238540, #72238542, …
maximal quotient of#72238530

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 109003);