C₂×#648363 #72331133

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴×C₄

Aliases

direct product

C₂×#648363

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

75899 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)(9,10)(11,12)(13,14)(15,16,18,20)(17,21,22,19);(7,8)(9,10)(12,13)(15,16,18,20)(17,19,22,21);(3,4)(5,6)(7,8)(9,10);(1,2);(4,6)(7,8,9,10)(11,13,14,12)(15,16)(17,21)(18,20)(19,22);(3,4)(5,6)(7,8)(9,10)(15,17)(16,19)(18,22)(20,21)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72261877, C₂×#72323283, C₂×#72263534, …
maximal normal subgroup of	C₂×#72261877, C₂×#72323283, C₂×#72263534, …

GAP

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

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

C2×#648363 #72331133