C₂×#687082 #72302651

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

Aliases

direct product

C₂×#687082

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

61635 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72280147, C₂×#72279977, C₂×#72280272, …
maximal normal subgroup of	C₂×#72280147, C₂×#72279977, C₂×#72280272, …
maximal quotient of	C₂×#72278330, C₂²×#687082

GAP

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

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

C2×#687082 #72302651