C₂×#4420586 #72295091

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴×C₄

Aliases

direct product

C₂×#4420586

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

82969 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)(13,17)(14,15)(16,18)(19,20)(21,23)(22,25)(24,26);(3,4,5,6)(7,8,9,10)(12,15)(13,18)(19,21)(20,23)(22,26)(24,25);(7,8)(9,10)(19,21,22,26)(20,24,25,23);(7,8)(9,10)(11,12)(13,16)(14,15)(17,18)(19,21)(20,24)(22,26)(23,25);(3,4)(5,6)(7,8)(9,10)(11,13)(12,16)(14,18)(15,17)(19,20)(21,24)(22,25)(23,26);(1,2)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293920, C₂×#72232034, C₂×#72232035, …
maximal normal subgroup of	C₂×#72293920, C₂×#72232034, C₂×#72232035, …
maximal quotient of	C₂×#72293990, C₂×#72250287, #72335071

GAP

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

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

C2×#4420586 #72295091