C₂×#7117 #124061

order	512
nilpotency class	4
isoclinism	isoclinic class of #1355
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#7117

Subgroups

	subgroup	G/H
center, socle	C₂³	#329
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂⁴	#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

562 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 42
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 42 points:

(1,2);(3,4)(5,9,10,7)(6,8)(11,12,14,19,15,20,28,38,16,21,27,39,30,18,35,36)(13,22,23,40,24,17,31,32,25,33,41,37,29,34,26,42);(3,5)(4,7)(6,10)(8,9)(11,13,16,25)(12,17,21,34)(14,26,27,31)(15,29,30,24)(18,33,20,22)(19,37,39,40)(23,28,41,35)(32,36,42,38)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 31257);