C₂×#13617 #512687

order	512
nilpotency class	5
isoclinism	isoclinic class of #2263
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

direct product

C₂×#13617

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×D₁₆
commutator	C₁₆	C₂³×C₄
Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	C₂³×D₁₆	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

902 conjugacy classes of subgroups; only the 282 normal subgroups are shown.

normal subgroups only

This group has 902 conjugacy classes of subgroups — too many to draw. Only its 282 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) = 38
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 38 points:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#151969, C₂×C₄×#2263, C₂×#152454, …
maximal normal subgroup of	C₂×#151969, C₂×C₄×#2263, C₂×#152454, …
maximal quotient of	C₂×#150058, #72286918, #72322606, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419883);