Q₈×#126 #36550

order	256
nilpotency class	2
isoclinism	isoclinic class of D₄×#120
automorphism	Aut(G) order 589824
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

Q₈×#126

Subgroups

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

852 conjugacy classes of subgroups; only the 497 normal subgroups are shown.

normal subgroups only

This group has 852 conjugacy classes of subgroups — too many to draw. Only its 497 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) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 24 points:

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

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	Q₈×#126⋊C₃, SL(2,𝔽₃)×#126, #11709992
maximal subgroup of	SD₁₆×#126, Q₁₆×#126, #6579593, …
maximal normal subgroup of	SD₁₆×#126, Q₁₆×#126, #6579593, …
maximal quotient of	Q₈×#388, Q₈×#387, C₄⋊C₄×#126, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 29538);