C₂×#1393523 #72331849

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂×#1393523

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

38223 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:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323300, C₂×#72323332
maximal normal subgroup of	C₂×#72323300, C₂×#72323332

GAP

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

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

C2×#1393523 #72331849