C₂×#13602 #512632

order	512
nilpotency class	5
isoclinism	isoclinic class of #1414
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#13602

Subgroups

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

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

normal subgroups only

This group has 850 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)(5,15,18,25)(6,11)(7,22)(8,13,24,28)(9,14)(10,17,26,34)(12,23,30,21)(16,31,33,27)(19,32,20,29);(3,5,9,20)(4,10)(6,16,7,17)(8,18,24,19)(11,25)(12,29)(13,31)(14,27)(15,30)(21,33,23,34)(22,32)(26,28);(3,6,9,7)(4,11,14,22)(5,16,20,17)(8,21,24,23)(10,25,27,32)(12,28,30,13)(15,31,29,26)(18,33,19,34)

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419828);