C₂×#516741 #72334706

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁴×C₄

Aliases

direct product

C₂×#516741

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

71185 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:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72334349, C₂×#72334350, C₂×#72309280, …
maximal normal subgroup of	C₂×#72334349, C₂×#72334350, C₂×#72309280, …
maximal quotient of	C₂×#72334408, C₂×#72335149

GAP

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

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

C2×#516741 #72334706