D₈×#442 #72279272

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

direct product

D₈×#442

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×D₄²
commutator	C₄²	C₂⁴×C₄
Frattini, cosocle	C₂×C₄²	C₂⁵
prime radical	C₂×D₈²	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

5212 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 20 points:

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

Conjugacy classes & character table

154 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,ℤ/8)×#442, #72280239, #72309590, …
maximal normal subgroup of	AGL(1,ℤ/8)×#442, #72280239, #72309590, …
maximal quotient of	#72309435, D₈×AGL(1,ℤ/16)

GAP

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

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

D8×#442 #72279272