C₂×#784590 #72302504

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂×#784590

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

39807 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:

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72280257, C₂×#72280033, C₂×#72280123, …
maximal normal subgroup of	C₂×#72280257, C₂×#72280033, C₂×#72280123, …

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

C2×#784590 #72302504