C₂×#3071227 #72294997

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂×#3071227

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

44281 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:

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

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293917, C₂×#72293918
maximal normal subgroup of	C₂×#72293917, C₂×#72293918
maximal quotient of	C₂×#72293939, C₂×#72293963

GAP

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

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

C2×#3071227 #72294997