C₂×#612437 #72334518

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

Aliases

direct product

C₂×#612437

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2410	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

74227 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:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×D₄×#2849, C₂×#72330175, C₂×#72330145
maximal normal subgroup of	C₂×D₄×#2849, C₂×#72330175, C₂×#72330145
maximal quotient of	C₂×#72332555

GAP

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

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

C2×#612437 #72334518