C₂×#13604 #512634

order	512
nilpotency class	5
isoclinism	isoclinic class of #13603
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#13604

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#354
commutator	C₄²	C₂³×C₄
Frattini, cosocle	#124	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#532 ◁ G
LECS	G ▷ #124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

840 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 194 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) = 34
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 34 points:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#512649, C₂×#152128, C₂×#152089
maximal normal subgroup of	C₂×#512649, C₂×#152128, C₂×#152089
maximal quotient of	#72293203, #72293207, #72293215

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419830);