C₂×#613508 #72339955

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁷

Aliases

direct product

C₂×#613508

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁵×#41	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

247301 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)(15,16)(17,22)(18,21)(19,20);(15,17,20,18)(16,21,19,22);(4,6)(8,10)(12,14)(15,18,20,17)(16,21,19,22);(8,10);(1,2);(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,19,20,16)(17,21,18,22)

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72337230
maximal normal subgroup of	C₂×#72337230
maximal quotient of	#72339991, #72339993, C₂×#72337385

GAP

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

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

C2×#613508 #72339955