C₂²×#1414 #512631

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

Aliases

direct product

C₂²×#1414

Subgroups

	subgroup	G/H
center, socle	C₂³	#354
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

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

normal subgroups only

This group has 894 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)(8,19,21,35)(9,23,25,13)(10,14)(11,16,26,33)(12,22)(18,32,24,31)(20,27,34,28)(29,36);(5,7)(6,11,14,28)(8,20)(9,24)(10,18)(12,15,29,32)(13,26,23,27)(16,21)(17,25)(19,30,35,31)(22,34)(33,36)

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, 419827);