Finite Group Databasebeta

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Cayley diagram

C2×#23993 #6451780

order512
nilpotency class3
isoclinismisoclinic class of #23990
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C210

Aliases

direct productC2×#23993

Subgroups

subgroupG/H
center, socleC22#2404
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3021C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

3854 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 34
transitive degreeμt(G) = 64

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

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6499858, #72306505, #72306516
maximal normal subgroup ofC2×#6499858, #72306505, #72306516

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6358976);