Finite Group Databasebeta

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Cayley diagram

C2×#9511 #201806

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

Aliases

direct productC2×#9511

Subgroups

subgroupG/H
center, socleC23#386
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#2404C2
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

2578 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)(5,7,9,10)(6,8)(11,12,15,19,16,18,25,13)(14,21,23,26,22,17,24,20);(3,5)(4,7,8,10)(6,9)(11,13,15,18,16,19,25,12)(14,20,24,21,22,26,23,17);(11,14)(12,17)(13,20)(15,24)(16,22)(18,21)(19,26)(23,25)

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#201819, #72235210, #72235212, …
maximal normal subgroup ofC2×#201819, #72235210, #72235212, …
maximal quotient of#72235800

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 109002);