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Cayley diagram

C2×#8537 #187954

order512
nilpotency class3
isoclinismisoclinic class of #8534
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C28

Aliases

direct productC2×#8537

Subgroups

subgroupG/H
center, socleC23#379
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

2546 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);(7,10)(11,12,14,18,15,17,23,25)(13,19,20,26,21,16,22,24);(3,4)(5,7)(6,8)(9,10)(11,13,15,21)(12,16,17,19)(14,22,23,20)(18,26,25,24);(3,5)(4,7)(6,9)(8,10)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#187956
maximal normal subgroup ofC2×#187956

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95150);