C₂×#72277937 #72297443

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

Aliases

direct product

C₂×#72277937

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 23038 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:

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

C2×#72277937 #72297443