C₂×#3464137 #72333101

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

direct product

C₂×#3464137

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

40025 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:

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

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323395, C₂×#72323406, C₂×#72323388
maximal normal subgroup of	C₂×#72323395, C₂×#72323406, C₂×#72323388

GAP

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

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

C2×#3464137 #72333101