C₂×#647970 #72311223

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

Aliases

direct product

C₂×#647970

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

58845 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72249699, C₂×#72268523, C₂×#72268529
maximal normal subgroup of	C₂×#72249699, C₂×#72268523, C₂×#72268529

GAP

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

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

C2×#647970 #72311223