C₂×#1344483 #72332886

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

Aliases

direct product

C₂×#1344483

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

31955 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)(11,12)(13,15)(14,17)(16,18)(20,23)(21,25);(7,8)(9,10)(14,18)(16,17)(19,20)(21,24)(22,23)(25,26);(12,13)(16,17)(19,20,22,23)(21,24,25,26);(1,2);(4,6)(11,13,15,12)(14,17,18,16)(19,20)(21,26)(22,23)(24,25);(3,4)(5,6)(11,14,15,18)(12,16,13,17)(19,21,22,25)(20,24,23,26)

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323415, C₂×#72323402, C₂×#72323409
maximal normal subgroup of	C₂×#72323415, C₂×#72323402, C₂×#72323409

GAP

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

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

C2×#1344483 #72332886