Finite Group Databasebeta

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Cayley diagram

C2×#19176 #6353650

order512
nilpotency class3
isoclinismisoclinic class of #19162
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28

Aliases

direct productC2×#19176

Subgroups

subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC42C25
prime radicalC2×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

1776 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 34
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 5 generators on 34 points:

(1,2);(3,4)(5,12)(6,10)(7,16)(8,11)(9,13)(14,17)(15,18)(19,20)(21,27)(22,24)(23,31)(25,29)(26,30)(28,34)(32,33);(3,5,7,13,8,14,6,15)(4,9,10,12,11,18,16,17)(20,24)(25,33)(26,31)(27,34);(19,21,22,28)(20,25,24,33)(23,32,30,29)(26,34,31,27);(3,6,8,7)(4,10,11,16)(5,13,14,15)(9,17,18,12)

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6260846);