C₂²×#8939 #72319618

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂¹²

Aliases

direct product

C₂²×#8939

Subgroups

	subgroup	G/H
center	C₂⁴×C₄	C₂×D₄
commutator	C₂×C₄	C₂⁴×C₈
Frattini, cosocle	C₂×C₄²	C₂⁵
socle	C₂⁵	C₂×C₂²⋊C₄
prime radical	C₂³×#121	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄ ◁ C₂⁴×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

17385 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

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

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

Conjugacy classes & character table

304 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂²×#208650
maximal normal subgroup of	C₂²×#208650
maximal quotient of	C₂²×#136613

GAP

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

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

C22×#8939 #72319618