C₂²×#12818 #72313797

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

direct product

C₂²×#12818

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂×#102
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂⁴×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ C₂³×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

9813 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 24 points:

(6,8)(9,10,11,16,14,18,13,23)(12,20,19,24,22,15,21,17);(1,2);(3,4);(1,2)(3,4)(10,15)(11,13)(12,19)(16,24)(17,23)(18,20)(21,22);(1,2)(3,4)(5,6,7,8)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#497864, C₂²×#495843, C₂²×D₄×#361
maximal normal subgroup of	C₂²×#497864, C₂²×#495843, C₂²×D₄×#361

GAP

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

gap> G := Group((6,8)(9,10,11,16,14,18,13,23)(12,20,19,24,22,15,21,17), (1,2), (3,4), (1,2)(3,4)(10,15)(11,13)(12,19)(16,24)(17,23)(18,20)(21,22), (1,2)(3,4)(5,6,7,8));

C22×#12818 #72313797