C₂×#7513 #153429

order	512
nilpotency class	3
isoclinism	isoclinic class of #99
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃₂ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#7513

Subgroups

	subgroup	G/H
center	C₂×C₁₆	C₂²⋊C₄
commutator	C₂²	C₂²×C₃₂
Frattini, cosocle	C₂×C₃₂	C₂³
socle	C₂²	C₂²⋊C₃₂
prime radical	C₂⁴	C₃₂
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₃₂ ◁ G
LECS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

188 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 66
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 66 points:

(1,2);(3,4,5,11,6,12,16,29,7,13,17,30,19,32,38,51,8,14,18,31,20,33,39,52,22,34,40,53,41,54,58,64,9,10,15,25,21,26,35,45,23,27,36,46,42,48,55,60,24,28,37,47,43,49,56,61,44,50,57,62,59,63,65,66);(4,10)(5,15)(12,26)(13,27)(14,28)(16,35)(17,36)(18,37)(32,48)(33,49)(34,50)(38,55)(39,56)(40,57)(54,63)(58,65)

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60625);