C₂×#12067 #483588

order	512
nilpotency class	4
isoclinism	isoclinic class of #2180
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#12067

Subgroups

	subgroup	G/H
center, socle	C₂³	#442
commutator	C₈	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂²×D₁₆	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

578 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 226 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) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 26 points:

(1,2);(3,4,6,9,7,5,8,10)(11,12,14,17,15,20,25,19)(13,21,24,18,23,16,22,26);(12,16)(13,22)(14,25)(17,18)(19,26)(20,21)(23,24);(3,5,6,10,7,4,8,9)(12,17,20,19)(13,23)(16,18,21,26)(22,24)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216174, C₂×AGL(1,ℤ/32), #72216330, …
maximal normal subgroup of	#72216174, C₂×AGL(1,ℤ/32), #72216330, …
maximal quotient of	#72216174, C₂×AGL(1,ℤ/32), #72216334, …
automorphism group of	#1417, C₃×SD₆₄

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390784);