C₂×#1393347 #72302825

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

Aliases

direct product

C₂×#1393347

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

28815 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)(21,24)(22,25)(23,26);(7,8,9,10)(13,17)(16,18)(19,21)(20,24)(22,23)(25,26);(7,8)(9,10)(12,15)(16,18)(19,22,24,26)(20,25,21,23);(4,6)(13,17)(16,18)(19,23,24,25)(20,26,21,22);(11,12,14,15)(13,18,17,16)(19,20,24,21)(22,23,26,25);(3,4)(5,6)(7,8)(9,10)(11,13)(12,16)(14,17)(15,18)(19,23)(20,22)(21,26)(24,25)

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72280482, C₂×#72280180, C₂×#72280272
maximal normal subgroup of	C₂×#72280482, C₂×#72280180, 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)(21,24)(22,25)(23,26), (7,8,9,10)(13,17)(16,18)(19,21)(20,24)(22,23)(25,26), (7,8)(9,10)(12,15)(16,18)(19,22,24,26)(20,25,21,23), (4,6)(13,17)(16,18)(19,23,24,25)(20,26,21,22), (11,12,14,15)(13,18,17,16)(19,20,24,21)(22,23,26,25), (3,4)(5,6)(7,8)(9,10)(11,13)(12,16)(14,17)(15,18)(19,23)(20,22)(21,26)(24,25));

C2×#1393347 #72302825