C₂×#688977 #72324408

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵

Aliases

direct product

C₂×#688977

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

55741 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:

(11,12)(13,18)(14,17)(15,16)(19,20)(21,25)(22,23)(24,26);(12,16)(13,17)(19,21)(20,24)(22,25)(23,26);(4,6)(8,10)(12,16)(14,18)(21,26)(24,25);(8,10)(11,13)(12,14)(15,17)(16,18)(19,22,23,20)(21,24,26,25);(1,2);(3,4)(5,6)(7,8)(9,10)(11,14)(12,17)(13,16)(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×#72232038, C₂×#72232060, C₂×#72232047, …
maximal normal subgroup of	C₂×#72232038, C₂×#72232060, C₂×#72232047, …

GAP

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

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

C2×#688977 #72324408