C₂×#647972 #72296289

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵

Aliases

direct product

C₂×#647972

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

60725 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232142, C₂×#72249699, C₂×#72232137, …
maximal normal subgroup of	C₂×#72232142, C₂×#72249699, C₂×#72232137, …
maximal quotient of	C₂×#72278117, C₂×#72277972, #72313014

GAP

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

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

C2×#647972 #72296289