C₂×#156824 #72334577

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

Aliases

direct product

C₂×#156824

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	C₂⁴×#121	C₂
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

74825 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 22 points:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72334303, C₂×#72298513, C₂×#72332588, …
maximal normal subgroup of	C₂×#72334303, C₂×#72298513, C₂×#72332588, …

GAP

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

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

C2×#156824 #72334577