C₂×#4930646 #72339954

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁷

Aliases

direct product

C₂×#4930646

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁴×H(5,𝔽₂)	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

248273 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72337230
maximal normal subgroup of	C₂×#72337230
maximal quotient of	#72339988, C₂×#72337387

GAP

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

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

C2×#4930646 #72339954