C₂×#617719 #72333137

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴×C₄

Aliases

direct product

C₂×#617719

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

88331 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323400, C₂×#72323406, C₂×#72323419, …
maximal normal subgroup of	C₂×#72323400, C₂×#72323406, C₂×#72323419, …
maximal quotient of	C₂×#72332554, C₂×#72323759

GAP

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

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

C2×#617719 #72333137