C₂×#646641 #72334469

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

Aliases

direct product

C₂×#646641

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

71765 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)(11,12,15,16)(13,17,18,14)(19,20)(21,22)(23,24)(25,26);(4,6)(8,10)(12,16)(14,17)(20,23)(21,26);(3,4)(5,6)(7,8)(9,10)(11,13,15,18)(12,14,16,17)(19,21)(20,25)(22,23)(24,26);(1,2);(8,10)(11,13)(12,17)(14,16)(15,18)(19,22)(20,21)(23,26)(24,25);(3,4)(5,6)(7,8)(9,10)(11,14,15,17)(12,18,16,13)(19,23,24,20)(21,25,26,22)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72233702, C₂×#72237112, C₂×#72233703, …
maximal normal subgroup of	C₂×#72233702, C₂×#72237112, C₂×#72233703, …
maximal quotient of	C₂×#72333291, C₂×#72334392, C₂×#72334411

GAP

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

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

C2×#646641 #72334469