C₂²×#1328 #480042

order	512
nilpotency class	4
isoclinism	isoclinic class of AΣL(1,𝔽₁₆)
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂²×#1328

Subgroups

	subgroup	G/H
center	C₂²×C₄	#99
commutator	C₂³	C₂³×C₈
Frattini, cosocle	C₂×M₁₆	C₂⁴
socle	C₂³	#324
prime radical	C₂⁵	C₂×C₈
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1038 conjugacy classes of subgroups; only the 322 normal subgroups are shown.

normal subgroups only

This group has 1038 conjugacy classes of subgroups — too many to draw. Only its 322 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);(5,6,8,13,9,14,19,27,10,12,18,24,20,25,31,34)(7,11,16,21,17,23,30,32,15,22,28,26,29,33,36,35);(5,7)(6,11,12,22)(8,16)(9,17)(10,15)(13,26,24,21)(14,23,25,33)(18,28)(19,30)(20,29)(27,35,34,32)(31,36)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387238);