C₂×#4306379 #72323957

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

Aliases

direct product

C₂×#4306379

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

39715 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)(5,6)(12,15)(13,17)(19,20,23,25)(21,26,24,22);(8,10)(11,12)(13,16)(14,15)(17,18);(7,8)(9,10)(19,21,23,24)(20,22,25,26);(4,5)(11,12)(13,18)(14,15)(16,17)(19,20)(21,22)(23,25)(24,26);(1,2);(3,5,6,4)(11,13)(12,16)(14,17)(15,18)(19,22)(20,24)(21,25)(23,26)

Conjugacy classes & character table

220 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323251, C₂×#72323261
maximal normal subgroup of	C₂×#72323251, C₂×#72323261

GAP

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

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

C2×#4306379 #72323957