C₂×#1259307 #72323900

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

Aliases

direct product

C₂×#1259307

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

50713 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:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323251, C₂×#72323247, C₂×#72323245, …
maximal normal subgroup of	C₂×#72323251, C₂×#72323247, C₂×#72323245, …

GAP

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

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

C2×#1259307 #72323900