C₂×#7059 #122259

order	512
nilpotency class	4
isoclinism	isoclinic class of #1343
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#7059

Subgroups

	subgroup	G/H
center	C₂²×C₄	#99
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₂²⋊C₈	C₂³
socle	C₂³	#323
prime radical	C₂⁵	M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂²×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₂²⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 472 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) = 50
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 50 points:

(1,2);(3,4)(5,11)(6,14,15,9)(7,10)(8,12,16,18)(13,17)(19,20,23,28,24,29,38,46,25,30,39,47,22,27,36,44)(21,31,33,48,32,42,49,45,34,43,37,50,35,26,40,41);(3,5,7,13)(4,8,10,16)(6,12,15,18)(9,11,14,17)(19,21)(20,26)(22,35)(23,37)(24,32)(25,34)(27,43)(28,45)(29,31)(30,42)(33,39)(36,49)(38,40)(41,47)(44,48)(46,50)

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, 29455);