C₂×#522070 #72299342

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂×#522070

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2847	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

40145 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:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72263798, C₂×#72298083, C₂×#72298081
maximal normal subgroup of	C₂×#72263798, C₂×#72298083, C₂×#72298081

GAP

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

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

C2×#522070 #72299342