#41×#347 #72310210

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

#41×#347

Subgroups

	subgroup	G/H
center	C₄²	C₂²×C₄²
commutator	C₂×C₄	C₂³×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁵
socle	C₂²	C₂³×#97
prime radical	C₂²×#41	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₄² ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3302 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

220 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313235, #72314111, #72315568, …
maximal normal subgroup of	#72313235, #72314111, #72315568, …

GAP

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

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