C₂²×#1407 #512589

order	512
nilpotency class	5
isoclinism	isoclinic class of #1406
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

direct product

C₂²×#1407

Subgroups

	subgroup	G/H
center, socle	C₂³	#353
commutator	C₄²	C₂³×C₄
Frattini, cosocle	#124	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂²×#124 ◁ G
LECS	G ▷ #124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

1154 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1154 conjugacy classes of subgroups — too many to draw. Only its 202 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) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419785);