C₂×#617359 #72286852

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄²

Aliases

direct product

C₂×#617359

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

170129 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 22 points:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72256572, C₂×#72237468, C₂×#72232303, …
maximal normal subgroup of	C₂×#72256572, C₂×#72237468, C₂×#72232303, …
maximal quotient of	C₂×#72252085, C₂×#72286421, C₂×#72287889, …

GAP

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

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

C2×#617359 #72286852