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Cayley diagram

C2×#61813 #10577992

order512
nilpotency class3
isoclinismisoclinic class of #61609
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C213

Aliases

direct productC2×#61813

Subgroups

subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC2×C4C26
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#532 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

6380 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 34
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 6 generators on 34 points:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72283271, #72283733
maximal normal subgroup of#72283271, #72283733

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10485188);