C₂²×#12816 #72313715

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

Aliases

direct product

C₂²×#12816

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂×#102
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂³×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ C₂³×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

8549 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)(10,14,15,23)(11,17)(12,20)(16,18,21,24)(19,22);(1,2);(3,4);(1,2)(3,4)(9,10)(11,18)(12,21)(13,15)(14,22)(16,20)(17,23)(19,24);(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×#495837, C₂²×#497864, C₂²×D₄×#360
maximal normal subgroup of	C₂²×#495837, C₂²×#497864, C₂²×D₄×#360

GAP

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

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

C22×#12816 #72313715