C₂×#1294219 #72325556

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄²

Aliases

direct product

C₂×#1294219

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

53697 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)(12,13)(15,16)(17,19)(18,21)(20,22);(11,12,14,13)(16,17)(20,21);(8,9)(12,13)(16,17)(18,22);(4,6)(8,9)(12,13)(15,17,19,16)(18,20,22,21);(1,2);(3,4)(5,6)(7,9,10,8)(11,13,14,12)(15,18,19,22)(16,20,17,21)

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232154, C₂×#72232166, C₂×#72232185
maximal normal subgroup of	C₂×#72232154, C₂×#72232166, C₂×#72232185

GAP

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

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

C2×#1294219 #72325556