C₂²×#15268 #72296086

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

Aliases

direct product

C₂²×#15268

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁴×#120	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

173221 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:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#648042, C₂²×#648387, C₂²×#647698, …
maximal normal subgroup of	C₂²×#648042, C₂²×#648387, C₂²×#647698, …
maximal quotient of	C₂×#72296146

GAP

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

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

C22×#15268 #72296086