C₂×#19073 #6351485

order	512
nilpotency class	3
isoclinism	isoclinic class of #19072
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

direct product

C₂×#19073

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

4116 conjugacy classes of subgroups; only the 594 normal subgroups are shown.

normal subgroups only

This group has 4116 conjugacy classes of subgroups — too many to draw. Only its 594 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) = 18
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 18 points:

(1,2);(4,8)(5,6)(9,10)(12,16)(13,14)(17,18);(3,4)(5,9)(6,8)(7,10)(11,12)(13,16)(14,18)(15,17);(3,4)(5,8)(6,9)(7,10)(13,14)(16,18);(3,5,7,6)(4,9,10,8)(11,13,15,14)(12,16,17,18)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279329
maximal normal subgroup of	#72279329
maximal quotient of	C₂×#177861, #72279329, #72279338

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6258681);