C₂²×#1400 #512553

order	512
nilpotency class	3
isoclinism	isoclinic class of #99
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₆ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂²×#1400

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂²⋊C₄
commutator	C₂²	C₂³×C₁₆
Frattini, cosocle	C₂×C₁₆	C₂⁴
socle	C₂³	C₂²⋊C₁₆
prime radical	C₂⁵	C₁₆
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

830 conjugacy classes of subgroups; only the 362 normal subgroups are shown.

normal subgroups only

This group has 830 conjugacy classes of subgroups — too many to draw. Only its 362 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);(5,6,7,12,8,13,16,24,9,14,17,25,18,26,29,34,10,11,15,21,19,22,27,31,20,23,28,32,30,33,35,36);(6,11)(7,15)(13,22)(14,23)(16,27)(17,28)(26,33)(29,35)

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419749);