C₂³×#1426 #72319009

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂³×#1426

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#367
commutator	C₁₆	C₂⁴×C₄
Frattini, cosocle	C₂×C₁₆	C₂⁵
prime radical	C₂⁵	SD₃₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈ ◁ C₂⁴×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

2677 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);(7,8,10,15,23,30,34,37,13,9,18,19,12,16,25,31,14,17,26,32,28,20,33,22,11,21,24,35,29,27,38,36);(5,6);(7,9,11,17,14,21,13,8)(10,22,24,36,26,37,18,31)(12,27,23,30,29,16,28,20)(15,25,19,34,32,38,35,33)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂³×#13658
maximal normal subgroup of	C₂³×#13658

GAP

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

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

C23×#1426 #72319009