C₂²×#14534 #72296910

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³×C₄²

Aliases

direct product

C₂²×#14534

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂³×#392	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

74957 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:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#3333823, C₂²×#1337713, C₂²×#3858074
maximal normal subgroup of	C₂²×#3333823, C₂²×#1337713, C₂²×#3858074

GAP

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

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

C22×#14534 #72296910