C₂×#7421 #150247

order	512
nilpotency class	5
isoclinism	isoclinic class of #7418
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#7421

Subgroups

	subgroup	G/H
center, socle	C₂³	#357
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂⁴	SD₃₂
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

308 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 74
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 74 points:

(1,2);(3,4)(5,9,10,7)(6,8)(11,12,15,21,18,25,41,24)(13,26,28,60,31,63,39,62)(14,32,35,69,38,55,74,70)(16,42,40,22,17,44,34,23)(19,46,47,27,51,64,50,66)(20,52,54,37,57,72,56,36)(29,67,43,61,30,68,45,53)(33,58,71,48,65,59,73,49);(3,5)(4,7)(6,10)(8,9)(11,13,18,31)(12,19,25,51)(14,33,38,65)(15,39,41,28)(16,43,17,45)(20,53,57,61)(21,50,24,47)(22,58,23,59)(26,32,63,55)(27,36,66,37)(29,34,30,40)(35,73,74,71)(42,48,44,49)(46,52,64,72)(54,68,56,67)(60,70,62,69)

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57443);