C₂×#850505 #72331854

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵

Aliases

direct product

C₂×#850505

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

54173 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)(8,10)(11,12)(13,17)(14,15)(16,18)(20,22)(24,26);(7,8)(9,10)(21,25)(24,26);(11,13)(12,16)(14,17)(15,18)(19,20,23,22)(21,26,25,24);(1,2);(11,13)(12,17)(14,16)(15,18)(19,21)(20,24)(22,26)(23,25);(3,4)(5,6)(7,8)(9,10)(11,14,15,12)(13,17,18,16)(19,22,23,20)(21,24,25,26)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323334, C₂×#72323324, C₂×#72323326
maximal normal subgroup of	C₂×#72323334, C₂×#72323324, C₂×#72323326

GAP

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

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

C2×#850505 #72331854