C₂×#652347 #72325614

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

Aliases

direct product

C₂×#652347

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂×#14715	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

52169 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232095, C₂×#72232101, C₂×#72232186
maximal normal subgroup of	C₂×#72232095, C₂×#72232101, C₂×#72232186

GAP

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

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

C2×#652347 #72325614