C₂²×#1413 #512626

order	512
nilpotency class	5
isoclinism	isoclinic class of #1412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂²×#1413

Subgroups

	subgroup	G/H
center, socle	C₂³	#354
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₄⋊Q₈	C₂⁴
prime radical	C₂⁴	#99
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂²×C₄⋊Q₈ ◁ G
LECS	G ▷ C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 822 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);(5,6,8,13,10,15,22,32)(7,16,18,34,17,33,35,30)(9,14,24,31,25,12,21,29)(11,26,27,19,28,20,36,23);(5,7,10,17)(6,11)(8,20,22,26)(9,23,25,19)(12,16)(13,30)(14,33)(15,28)(18,21,35,24)(27,29)(31,36)(32,34)

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, 419822);