C₂×#847085 #72294015

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

Aliases

direct product

C₂×#847085

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

70255 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72233706, C₂×#72233693, C₂×#72233687, …
maximal normal subgroup of	C₂×#72233706, C₂×#72233693, C₂×#72233687, …
maximal quotient of	C₂×#72293943, C₂×#72334392, C₂×#72334396

GAP

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

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

C2×#847085 #72294015