C₂×#703996 #72302602

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

Aliases

direct product

C₂×#703996

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂³×#546	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

55149 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72280204, C₂×#72280430, C₂×#72280221
maximal normal subgroup of	C₂×#72280204, C₂×#72280430, C₂×#72280221

GAP

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

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

C2×#703996 #72302602