D₈×#360 #72310051

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

D₈×#360

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×#102
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	D₈×#135	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×D₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1710 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 24 points:

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

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72310057, D₈×AGL(1,ℤ/16), #72310128, …
maximal normal subgroup of	#72310057, D₈×AGL(1,ℤ/16), #72310128, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

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

D8×#360 #72310051