C₂×#846204 #72323948

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

Aliases

direct product

C₂×#846204

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

40617 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323249, C₂×#72323250, C₂×#72323262, …
maximal normal subgroup of	C₂×#72323249, C₂×#72323250, C₂×#72323262, …

GAP

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

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

C2×#846204 #72323948