C₂×#648357 #72341795

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

Aliases

direct product

C₂×#648357

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂³×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

57681 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)(5,6)(7,8)(9,10)(11,12,14,13)(15,16)(17,19)(18,21)(20,22);(3,4)(5,6)(7,8,9,10)(12,13)(15,17,18,22)(16,19,21,20);(4,6)(8,10);(11,13,14,12)(17,22)(19,20);(11,12,14,13)(15,17)(16,20)(18,22)(19,21);(3,4)(5,6)(7,8)(9,10)(15,16)(17,20)(18,21)(19,22)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72260984, C₂×#72281143, C₂×#120×C₂²⋊Q₈
maximal normal subgroup of	C₂×#72260984, C₂×#72281143, C₂×#120×C₂²⋊Q₈

GAP

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

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

C2×#648357 #72341795