C₂×#202442 #72321971

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

direct product

C₂×#202442

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂⁴×C₄
Frattini, cosocle	C₂×C₄²	C₂⁵
socle	C₂⁴	C₂×C₄×D₄
prime radical	C₂²×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

5881 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 30 points:

(4,6)(7,8,9,12)(10,11,14,13)(15,16,19,22,20,23,24,17)(18,26,28,25,27,21,29,30);(1,2);(8,11)(10,14)(12,13)(15,17,24,23,20,22,19,16)(18,25,29,26,27,30,28,21);(3,4,5,6)(8,12)(11,13)(15,18,24,29,20,27,19,28)(16,21,17,25,23,26,22,30);(18,27)(21,26)(25,30)(28,29)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72318378, C₂×#72318352, C₂×#72318426
maximal normal subgroup of	C₂×#72318378, C₂×#72318352, C₂×#72318426

GAP

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

gap> G := Group((4,6)(7,8,9,12)(10,11,14,13)(15,16,19,22,20,23,24,17)(18,26,28,25,27,21,29,30), (1,2), (8,11)(10,14)(12,13)(15,17,24,23,20,22,19,16)(18,25,29,26,27,30,28,21), (3,4,5,6)(8,12)(11,13)(15,18,24,29,20,27,19,28)(16,21,17,25,23,26,22,30), (18,27)(21,26)(25,30)(28,29));

C2×#202442 #72321971