C₂×#516323 #72267239

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁸

Aliases

direct product

C₂×#516323

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁴×H(5,𝔽₂)	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

263245 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:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×#72233412, C₂×#72246282, C₂×#72252244
maximal subgroup of	C₂×#72237497, C₂×#72233412, C₂×#72237500, …
maximal normal subgroup of	C₂×#72237497, C₂×#72233412, C₂×#72237500, …
maximal quotient of	C₂×#72298498, C₂×#72252090, #72335069, …

GAP

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

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

C2×#516323 #72267239