C₂×#626400 #72333859

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

Aliases

direct product

C₂×#626400

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

50073 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)(12,15)(13,17);(4,6)(7,8)(9,10)(12,15)(16,18)(19,20)(21,24)(22,25)(23,26);(19,21)(20,23)(22,26)(24,25);(1,2);(3,4,5,6)(11,12,14,15)(13,16,17,18)(19,21)(20,24)(22,26)(23,25);(11,13,14,17)(12,16,15,18)(19,20)(21,23)(22,25)(24,26)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72297845, C₂×#72327589, C₂×#72327602
maximal normal subgroup of	C₂×#72297845, C₂×#72327589, C₂×#72327602

GAP

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

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

C2×#626400 #72333859