D₈×#361 #72309883

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

Aliases

direct product

D₈×#361

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×#102
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	C₂×D₈²	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

1965 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:

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

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309890, D₈×AGL(1,ℤ/16), AGL(1,ℤ/8)×#361
maximal normal subgroup of	#72309890, D₈×AGL(1,ℤ/16), AGL(1,ℤ/8)×#361

GAP

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

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

D8×#361 #72309883