C₂×#535919 #72299485

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

Aliases

direct product

C₂×#535919

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2388	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

35625 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:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72298083, C₂×#72297195, C₂×#72298087
maximal normal subgroup of	C₂×#72298083, C₂×#72297195, C₂×#72298087

GAP

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

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

C2×#535919 #72299485