C₂×#5955445 #72334465

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

Aliases

direct product

C₂×#5955445

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

73297 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232405, C₂×#72237112, C₂×#72237462, …
maximal normal subgroup of	C₂×#72232405, C₂×#72237112, C₂×#72237462, …
maximal quotient of	C₂×#72333291, C₂×#72323618

GAP

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

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

C2×#5955445 #72334465