C₂²×#15258 #72268562

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁷×C₄

Aliases

direct product

C₂²×#15258

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

257749 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 20 points:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#647656, C₂²×#648360, C₂²×#846785, …
maximal normal subgroup of	C₂²×#647656, C₂²×#648360, C₂²×#846785, …
maximal quotient of	C₂×#72252085, C₂×#72278210, C₂²×#158317, …

GAP

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

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

C22×#15258 #72268562