C₂×#536491 #72299481

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

Aliases

direct product

C₂×#536491

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2431	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

37101 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 22 points:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72298095, C₂×#72298084, C₂×#72298087
maximal normal subgroup of	C₂×#72298095, C₂×#72298084, C₂×#72298087

GAP

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

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

C2×#536491 #72299481