C₂²×#15127 #72320544

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵×C₄

Aliases

direct product

C₂²×#15127

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁴×#127	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

101813 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 512

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

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#649052, C₂²×#616153, C₂²×#774397, …
maximal normal subgroup of	C₂²×#649052, C₂²×#616153, C₂²×#774397, …
maximal quotient of	C₂²×#158362

GAP

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

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

C22×#15127 #72320544