C₂×#824762 #72302828

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂×#824762

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

37853 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:

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72280449, C₂×#72280331, C₂×#72280272
maximal normal subgroup of	C₂×#72280449, C₂×#72280331, C₂×#72280272

GAP

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

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

C2×#824762 #72302828