C₂×#542281 #72332915

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

Aliases

direct product

C₂×#542281

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2669	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

36977 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:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323407, C₂×#72297536, C₂×#72323421
maximal normal subgroup of	C₂×#72323407, C₂×#72297536, C₂×#72323421

GAP

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

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

C2×#542281 #72332915