C₂×#945537 #72324935

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂×#945537

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

50545 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:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232077, C₂×#72232125, C₂×#72232157, …
maximal normal subgroup of	C₂×#72232077, C₂×#72232125, C₂×#72232157, …

GAP

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

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

C2×#945537 #72324935