C₂²×#1411 #512603

order	512
nilpotency class	5
isoclinism	isoclinic class of #1408
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

direct product

C₂²×#1411

Subgroups

	subgroup	G/H
center, socle	C₂³	#353
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

1102 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1102 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)(7,16,17,30,19,27,34,35)(9,14,24,31,25,12,21,29)(11,20,26,23,28,33,36,18);(5,7,10,19)(6,11,15,28)(8,20,22,33)(9,23,25,18)(12,27,14,16)(13,30,32,35)(17,21,34,24)(26,31,36,29)

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, 419799);