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Cayley diagram

C2×#33539 #7622845

order512
nilpotency class4
isoclinismisoclinic class of #33539
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C210

Aliases

direct productC2×#33539

Subgroups

subgroupG/H
center, socleC22C2×#447
commutator, Frattini, cosocleC2×C8C25
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

3978 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)(5,16)(6,20)(7,15)(8,14)(9,12)(10,17)(11,18)(13,23)(19,31)(21,32)(22,26)(24,30)(25,29)(27,33)(28,34);(3,5)(4,10)(6,21)(7,18)(8,25)(9,19)(11,26)(12,27)(13,29)(14,28)(15,32)(16,20)(17,23)(22,33)(24,34)(30,31);(3,5)(4,11)(6,17)(7,25)(8,18)(9,19)(10,26)(12,29)(13,27)(14,31)(15,16)(20,32)(21,23)(22,34)(24,33)(28,30);(4,12)(10,27)(11,29)(13,26)(14,15)(16,31)(20,30)(28,32)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72231505, C2×#495739, #72231549, …
maximal normal subgroup of#72231505, C2×#495739, #72231549, …
maximal quotient of#72249915, #72252745, #72264986, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530041);