C₂×#1409841 #72299201

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

Aliases

direct product

C₂×#1409841

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

36147 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 26 points:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72297586, C₂×#72297596
maximal normal subgroup of	C₂×#72297586, C₂×#72297596

GAP

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

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

C2×#1409841 #72299201