C₂×#789380 #72302689

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

Aliases

direct product

C₂×#789380

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

30417 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 26 points:

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72280075, C₂×#72279963, C₂×#72280272
maximal normal subgroup of	C₂×#72280075, C₂×#72279963, C₂×#72280272

GAP

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

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

C2×#789380 #72302689