C₂×#519363 #72341745

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁵

Aliases

direct product

C₂×#519363

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂³×#525	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

60441 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 22 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×C₂²⋊C₄×#525, C₂×#72303265, C₂×#72268529
maximal normal subgroup of	C₂×C₂²⋊C₄×#525, C₂×#72303265, C₂×#72268529

GAP

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

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

C2×#519363 #72341745