C₂×#517617 #72340061

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁷

Aliases

direct product

C₂×#517617

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁴×H(5,𝔽₂)	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

251077 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:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72337324
maximal normal subgroup of	C₂×#72337324
maximal quotient of	#72340074, C₂×#72337387

GAP

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

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

C2×#517617 #72340061