C₂×#7286 #127835

order	512
nilpotency class	4
isoclinism	isoclinic class of #1343
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₂×#7286

Subgroups

	subgroup	G/H
center	C₂²×C₄	#99
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂×C₄×Q₈	C₂³
socle	C₂³	#342
prime radical	C₂⁴	#95
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

606 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 606 conjugacy classes of subgroups — too many to draw. Only its 162 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);(3,4,6,11,8,12,7,10)(5,13,14,9,16,17,15,18)(20,24)(21,27)(25,33)(26,32)(28,29)(31,34);(3,5,7,15,8,16,6,14)(4,9,11,17,12,18,10,13)(19,20,22,25,23,26,30,34)(21,24,28,31,27,32,29,33)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35031);