C₂×#13652 #512758

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

Aliases

direct product

C₂×#13652

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×D₁₆
commutator	C₁₆	C₂³×C₄
Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

462 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 186 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) = 256

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

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₄×#2263, C₂×#151083
maximal normal subgroup of	C₂×C₄×#2263, C₂×#151083
maximal quotient of	C₂×#151078

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419954);