C₂×#641340 #72325367

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

direct product

C₂×#641340

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

53725 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)(8,9)(11,12)(13,18)(14,17)(15,16)(19,20,24,26)(21,25,22,23);(3,4,6,5)(7,8,10,9)(12,16)(13,17)(19,21)(20,23)(22,24)(25,26);(21,22)(23,25);(7,8,10,9)(11,13)(12,14)(15,17)(16,18)(19,22,24,21)(20,25,26,23);(1,2);(3,5,6,4)(7,9,10,8)(11,14)(12,17)(13,16)(15,18)(19,23,24,25)(20,21,26,22)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232096, C₂×#72232125, C₂×#72232149
maximal normal subgroup of	C₂×#72232096, C₂×#72232125, C₂×#72232149

GAP

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

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

C2×#641340 #72325367