C₂×#650493 #72324066

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂×#650493

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2408	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

41833 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323258, C₂×#72323263, C₂×#72262141
maximal normal subgroup of	C₂×#72323258, C₂×#72323263, C₂×#72262141

GAP

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

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

C2×#650493 #72324066