C₂×#7089 #122655

order	512
nilpotency class	4
isoclinism	isoclinic class of #7076
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#7089

Subgroups

	subgroup	G/H
center, socle	C₂³	#327
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄⋊C₈	C₂³
prime radical	C₂⁴	#102
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₈ ◁ G
LECS	G ▷ C₂×C₄⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

406 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 128

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

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

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, 29851);