#97×AGL(1,ℤ/8) #72279643

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

#97×AGL(1,ℤ/8)

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂³×D₄
commutator	C₂×C₄	C₂³×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁵
socle	C₂³	C₂²×C₄×D₄
prime radical	C₂²×AGL(1,ℤ/8)	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×#97 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

5187 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:

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

Conjugacy classes & character table

220 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,ℤ/8)×#442, #72309619, AGL(1,ℤ/8)×#347, …
maximal normal subgroup of	AGL(1,ℤ/8)×#442, #72309619, AGL(1,ℤ/8)×#347, …
maximal quotient of	AGL(1,ℤ/8)×#347

GAP

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

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