C₂×#1782141 #72332525

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

direct product

C₂×#1782141

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

41317 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)(11,12,14,16)(13,15,18,17)(19,20,22,25)(21,24,26,23);(7,8)(9,10)(19,21)(20,23)(22,26)(24,25);(4,6)(11,12,14,16)(13,17,18,15)(19,21,22,26)(20,24,25,23);(1,2);(3,4,5,6)(11,12)(13,17)(14,16)(15,18)(19,21)(20,24)(22,26)(23,25);(11,13)(12,15)(14,18)(16,17)(19,20)(21,23)(22,25)(24,26)

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323385, C₂×#72323415, C₂×#72323395
maximal normal subgroup of	C₂×#72323385, C₂×#72323415, C₂×#72323395

GAP

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

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

C2×#1782141 #72332525