C₂×#7832 #165814

order	512
nilpotency class	3
isoclinism	isoclinic class of #7734
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#7832

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

783 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 783 conjugacy classes of subgroups — too many to draw. Only its 194 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) = 50
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 50 points:

(1,2);(3,4,8,11)(5,12,14,17)(6,10,7,15)(9,16,18,13)(19,20,23,26)(21,24,28,31)(22,25,29,32)(27,33,34,30)(35,36,39,43)(37,40,45,49)(38,41,46,42)(44,47,50,48);(3,5,8,14)(4,9,11,18)(6,13,7,16)(10,17,15,12)(19,21)(20,24)(22,27)(23,28)(25,30)(26,31)(29,34)(32,33)(35,37,39,45)(36,40,43,49)(38,44,46,50)(41,47,42,48);(3,6,8,7)(4,10,11,15)(5,13,14,16)(9,17,18,12)(19,22)(20,25)(21,27)(23,29)(24,30)(26,32)(28,34)(31,33)(36,41)(38,46)(40,47)(42,43)(44,50)(48,49)

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 73010);