C₂×#4034097 #72332650

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂×#4034097

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

42873 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 26 points:

(8,10)(11,12)(13,16)(14,15)(17,18)(20,23)(24,25);(3,4)(5,6)(7,8)(9,10)(11,13,15,17)(12,16,14,18);(12,14)(16,18)(19,20,22,23)(21,25,26,24);(1,2);(4,6)(11,14,15,12)(13,16,17,18)(19,20)(21,24)(22,23)(25,26);(3,4)(5,6)(11,13)(12,16)(14,18)(15,17)(19,21)(20,24)(22,26)(23,25)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323418, C₂×#72323402, C₂×#72323410
maximal normal subgroup of	C₂×#72323418, C₂×#72323402, C₂×#72323410

GAP

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

gap> G := Group((8,10)(11,12)(13,16)(14,15)(17,18)(20,23)(24,25), (3,4)(5,6)(7,8)(9,10)(11,13,15,17)(12,16,14,18), (12,14)(16,18)(19,20,22,23)(21,25,26,24), (1,2), (4,6)(11,14,15,12)(13,16,17,18)(19,20)(21,24)(22,23)(25,26), (3,4)(5,6)(11,13)(12,16)(14,18)(15,17)(19,21)(20,24)(22,26)(23,25));

C2×#4034097 #72332650