Q₈×#383 #520520

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄×#120
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

Q₈×#383

Subgroups

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

2514 conjugacy classes of subgroups; only the 1178 normal subgroups are shown.

normal subgroups only

This group has 2514 conjugacy classes of subgroups — too many to draw. Only its 1178 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) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 28 points:

(21,22,24,25)(23,27,28,26);(21,23,24,28)(22,26,25,27);(1,2,3,4)(7,11)(9,12)(14,17)(15,19)(21,25,24,22)(23,27,28,26);(5,6,8,10)(7,9,11,12)(13,14)(15,18)(16,17)(19,20);(5,7)(6,9)(8,11)(10,12)(13,15,16,19)(14,18,17,20)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 427716);