C₂³×#7389 #72318962

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂¹⁰

Aliases

direct product

C₂³×#7389

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#1333
commutator	C₁₆	C₂⁴×C₈
Frattini, cosocle	#346	C₂⁵
prime radical	C₂³×#508	C₄
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ C₂³×C₄×C₈ ◁ C₂³×#346 ◁ G
LECS	G ▷ #346 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

10184 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 38
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 38 points:

(1,2);(3,4);(8,13,14,27,15,28,25,30)(9,17,19,35)(10,21)(11,24)(16,33,31,29,32,18,22,37)(20,26,38,36)(23,34);(5,6);(7,8)(9,18)(10,22)(11,25)(12,15)(13,26)(14,24)(16,34)(17,27)(19,33)(20,37)(21,32)(23,31)(28,36)(29,38)(30,35)

Conjugacy classes & character table

272 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂³×AGL(1,ℤ/32)
maximal normal subgroup of	C₂³×AGL(1,ℤ/32)

GAP

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

gap> G := Group((1,2), (3,4), (8,13,14,27,15,28,25,30)(9,17,19,35)(10,21)(11,24)(16,33,31,29,32,18,22,37)(20,26,38,36)(23,34), (5,6), (7,8)(9,18)(10,22)(11,25)(12,15)(13,26)(14,24)(16,34)(17,27)(19,33)(20,37)(21,32)(23,31)(28,36)(29,38)(30,35));

C23×#7389 #72318962