C₂²×#1358 #483030

order	512
nilpotency class	4
isoclinism	isoclinic class of #1357
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂²×#1358

Subgroups

	subgroup	G/H
center, socle	C₂³	#331
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂⁴	#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1082 conjugacy classes of subgroups; only the 322 normal subgroups are shown.

normal subgroups only

This group has 1082 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);(5,6,8,13,9,12,21,28,10,15,22,31,24,14,32,30)(7,16,17,33,18,27,34,36,19,26,35,29,23,11,20,25);(5,7,10,19)(6,11,15,27)(8,20,22,34)(9,23,24,18)(12,26,14,16)(13,29,31,33)(17,21,35,32)(25,28,36,30)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390226);