C₂×#687721 #72299298

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

Aliases

direct product

C₂×#687721

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

38457 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72298098, C₂×#72298085, C₂×#72298080
maximal normal subgroup of	C₂×#72298098, C₂×#72298085, C₂×#72298080

GAP

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

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

C2×#687721 #72299298