C₂×#516588 #72334540

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁵

Aliases

direct product

C₂×#516588

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

74797 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:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72330175, C₂×C₄×#34980
maximal normal subgroup of	C₂×#72330175, C₂×C₄×#34980
maximal quotient of	C₂×#72317651

GAP

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

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

C2×#516588 #72334540