C₂×#996114 #72294779

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

Aliases

direct product

C₂×#996114

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

53115 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)(12,14)(15,16,18,21)(17,19,22,20);(4,6)(8,10)(15,17,18,22)(16,19,21,20);(4,6)(11,12)(13,14)(15,17,18,22)(16,20,21,19);(7,8)(9,10)(15,16)(17,20)(18,21)(19,22);(3,4)(5,6)(7,8)(9,10)(15,17)(16,20)(18,22)(19,21);(1,2)

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293946
maximal normal subgroup of	C₂×#72293946
maximal quotient of	C₂×#72293968

GAP

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

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

C2×#996114 #72294779