C₂×D₁₆² #72225592

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

direct product

C₂×D₁₆²

Subgroups

	subgroup	G/H
center, socle	C₂³	D₈²
commutator, Frattini, cosocle	C₈²	C₂⁵
LCS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

4976 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 34
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 34 points:

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

Conjugacy classes & character table

242 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226981, #72227092, #72227114
maximal normal subgroup of	#72226981, #72227092, #72227114
maximal quotient of	#72225593

GAP

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

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

C2×D162 #72225592