Finite Group Databasebeta

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Cayley diagram

C2×#10818 #353527

order512
nilpotency class3
isoclinismisoclinic class of #465
automorphism?
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4

Aliases

direct productC2×#10818

Subgroups

subgroupG/H
centerC23×C4C2×D4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
socleC24C4×D4
prime radicalC25#41
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C4 ◁ C24×C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

2114 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)(5,6)(7,8,9,10)(11,12)(13,21)(14,17)(15,19)(16,20)(18,23)(22,26)(24,25);(12,17,20,19)(18,21,26,25);(4,6)(11,13,14,22,16,24,15,23)(12,18,19,25,20,26,17,21)

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72215705, C2×#378848, #72216085
maximal normal subgroup of#72215705, C2×#378848, #72216085
maximal quotient of#72215665, C2×#108578

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260723);