C₂×#516581 #72334360

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

Aliases

direct product

C₂×#516581

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂³×#535	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

90485 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)(15,16,19,20)(17,18,21,22);(4,6)(7,8,9,10)(11,12,13,14)(17,21)(18,22);(3,4)(5,6)(7,8)(9,10)(16,20)(17,21);(3,4)(5,6)(7,8)(9,10)(15,17)(16,18)(19,21)(20,22);(1,2);(15,18)(16,17)(19,22)(20,21)

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72256584, C₂×#72256581, C₂×#72256586, …
maximal normal subgroup of	C₂×#72256584, C₂×#72256581, C₂×#72256586, …
maximal quotient of	C₂×#72298505, C₂×#72298440

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

C2×#516581 #72334360