C₂×#991126 #72325408

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

Aliases

direct product

C₂×#991126

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

64419 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 18 points:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232101, C₂×#72232156, C₂×#72232110
maximal normal subgroup of	C₂×#72232101, C₂×#72232156, C₂×#72232110

GAP

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

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

C2×#991126 #72325408