C₂×#536667 #72296434

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂×#536667

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2403	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

55553 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)(9,10)(12,14)(15,16)(17,21)(18,22)(19,20);(8,9)(11,12)(13,14)(15,17)(16,18)(19,22)(20,21);(8,9)(16,20)(17,22);(8,9)(15,18)(16,17)(19,21)(20,22);(3,4,5,6)(7,9,10,8)(15,17)(16,18)(19,22)(20,21);(1,2)

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232174, C₂×#72232085, C₂×#72263798, …
maximal normal subgroup of	C₂×#72232174, C₂×#72232085, C₂×#72263798, …
maximal quotient of	C₂×#72278184

GAP

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

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

C2×#536667 #72296434