C₂×#2601324 #72325526

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁰×C₄³

Aliases

direct product

C₂×#2601324

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

44365 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:

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

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232181, C₂×#72232154, C₂×#72232147
maximal normal subgroup of	C₂×#72232181, C₂×#72232154, C₂×#72232147

GAP

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

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

C2×#2601324 #72325526