C₂×#2111455 #72337858

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

direct product

C₂×#2111455

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

41773 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:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232279, C₂×#72232336, C₂×#72232296
maximal normal subgroup of	C₂×#72232279, C₂×#72232336, C₂×#72232296

GAP

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

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

C2×#2111455 #72337858