C₂×#72264988 #72296474

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁷×C₄

Aliases

direct product

C₂×#72264988

Subgroups

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

At least 10134 conjugacy classes of subgroups; only the 9184 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 26 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72278229
maximal normal subgroup of	C₂×#72278229

GAP

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

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

C2×#72264988 #72296474