C₂×#648612 #72337929

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂×#648612

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2433	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

53435 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232305, C₂×#72232311, C₂×#72232346
maximal normal subgroup of	C₂×#72232305, C₂×#72232311, C₂×#72232346

GAP

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

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

C2×#648612 #72337929