C₂×#2032043 #72332504

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂×#2032043

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

47355 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:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323386, C₂×#72323397, C₂×#72263091
maximal normal subgroup of	C₂×#72323386, C₂×#72323397, C₂×#72263091

GAP

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

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

C2×#2032043 #72332504