C₂×#7751 #165682

order	512
nilpotency class	3
isoclinism	isoclinic class of #7734
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#7751

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

1337 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1337 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 — 4 generators on 34 points:

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

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, 72878);