C₂×#7046 #96190

order	512
nilpotency class	3
isoclinism	isoclinic class of #7035
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#7046

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁵	C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

942 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 942 conjugacy classes of subgroups — too many to draw. Only its 202 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) = 256

A minimal faithful permutation representation — 3 generators on 34 points:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3386);