C₂×#687598 #72323708

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵×C₄

Aliases

direct product

C₂×#687598

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

80509 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323733, C₂×#72215608, C₂×#72232055, …
maximal normal subgroup of	#72323733, C₂×#72215608, C₂×#72232055, …

GAP

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

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

C2×#687598 #72323708