C₂×#876609 #72296376

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

Aliases

direct product

C₂×#876609

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

42081 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72278001, C₂×#72323334, C₂×#72323314
maximal normal subgroup of	C₂×#72278001, C₂×#72323334, C₂×#72323314
maximal quotient of	C₂×#72278125

GAP

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

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

C2×#876609 #72296376