C₂×#846755 #72332868

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

Aliases

direct product

C₂×#846755

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

49171 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)(19,20)(21,26)(22,25)(23,24);(7,8)(9,10)(12,15)(16,18)(19,21)(20,22)(23,26)(24,25);(19,22)(20,25)(21,23)(24,26);(1,2);(4,6)(11,12,14,15)(13,16,17,18)(19,22)(20,21)(23,25)(24,26);(3,4)(5,6)(11,13,14,17)(12,16,15,18)(19,23,24,20)(21,26,25,22)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323394, C₂×#72297762, C₂×#72323403
maximal normal subgroup of	C₂×#72323394, C₂×#72297762, C₂×#72323403

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

C2×#846755 #72332868