C₂×#30237 #6435887

order	512
nilpotency class	3
isoclinism	isoclinic class of #30236
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂×#30237

Subgroups

	subgroup	G/H
center, socle	C₂²	#2644
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1208 conjugacy classes of subgroups; only the 490 normal subgroups are shown.

normal subgroups only

This group has 1208 conjugacy classes of subgroups — too many to draw. Only its 490 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) = 34
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 34 points:

(1,2);(3,4,8,22,9,11,7,13)(5,14,17,25,18,23,15,10)(6,12,20,27,21,26,19,28)(16,29,31,34,32,33,30,24);(3,5)(4,10,11,25)(6,16)(7,17)(8,15)(9,18)(12,24,26,34)(13,14,22,23)(19,31)(20,30)(21,32)(27,33,28,29);(4,11)(5,15)(6,19)(7,8)(10,23)(12,27)(14,25)(17,18)(20,21)(26,28)(29,33)(30,31);(3,6,8,20,9,21,7,19)(4,12,22,27,11,26,13,28)(5,16,15,30,18,32,17,31)(10,24,23,33,25,34,14,29)

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6343083);