C₂×#696084 #72302597

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄²

Aliases

direct product

C₂×#696084

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

55415 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72280357, C₂×#72280257, C₂×#72280535, …
maximal normal subgroup of	C₂×#72280357, C₂×#72280257, C₂×#72280535, …

GAP

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

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

C2×#696084 #72302597