C₂×#524461 #72298641

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁶

Aliases

direct product

C₂×#524461

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂³×D₄²	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

86177 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)(12,14)(16,19)(20,22);(3,4,6,5)(7,8)(9,10)(11,12)(13,14);(8,10)(17,21)(20,22);(3,4,6,5)(7,8)(9,10)(15,16)(17,20)(18,19)(21,22);(3,5,6,4)(7,8)(9,10)(15,17)(16,20)(18,21)(19,22);(1,2)

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232081, C₂×#72233558, C₂×#72233563, …
maximal normal subgroup of	C₂×#72232081, C₂×#72233558, C₂×#72233563, …
maximal quotient of	#72323732, C₂²×#524461, #72330612, …

GAP

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

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

C2×#524461 #72298641