Finite Group Databasebeta

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Cayley diagram

C2×#10321 #234281

order512
nilpotency class3
isoclinismisoclinic class of #10320
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C27

Aliases

direct productC2×#10321

Subgroups

subgroupG/H
center, socleC23#392
commutator, Frattini, cosocleC23×C4C24
prime radicalC2×#3186C2
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

2080 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 4 generators on 26 points:

(1,2);(3,4)(5,7)(6,9)(8,10)(11,12)(13,16)(14,17)(15,18)(19,24)(20,23)(21,25)(22,26);(4,7)(5,10)(8,9)(11,13)(12,16)(14,19)(15,20)(17,24)(18,23)(21,26)(22,25);(11,14,15,21)(12,17,18,25)(13,19,20,26)(16,22,23,24)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72257298, #72257318, C2×#234329
maximal normal subgroup of#72257298, #72257318, C2×#234329
maximal quotient of#72264628

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 141477);