C₂×#72277898 #72297335

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂×#72277898

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁶
commutator, Frattini, cosocle	C₂⁴	C₂⁷
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

At least 23066 conjugacy classes of subgroups; only the 35280 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 26 points:

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

Conjugacy classes & character table

272 conjugacy classes. The character table is too large to display.

Relationships

maximal quotient of

C₂×#72278177

GAP

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

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

C2×#72277898 #72297335