Q₈×#125 #36549

order	256
nilpotency class	2
isoclinism	isoclinic class of D₄×#120
automorphism	Aut(G) order 786432
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

Q₈×#125

Subgroups

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

831 conjugacy classes of subgroups; only the 543 normal subgroups are shown.

normal subgroups only

This group has 831 conjugacy classes of subgroups — too many to draw. Only its 543 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) = 256

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

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

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	SL(2,𝔽₃)×#125
maximal subgroup of	#98280, Q₈×#330, #224171, …
maximal normal subgroup of	#98280, Q₈×#330, #224171, …
maximal quotient of	Q₈×#387, Q₈×#389, C₄⋊C₄×#125, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 29537);