C₂×#652047 #72299308

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

Aliases

direct product

C₂×#652047

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2537	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

27801 conjugacy classes of subgroups.

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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72297416, C₂×#72297597, C₂×#72297560
maximal normal subgroup of	C₂×#72297416, C₂×#72297597, C₂×#72297560

GAP

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

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

C2×#652047 #72299308