C₂×#3071131 #72294538

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

Aliases

direct product

C₂×#3071131

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

40767 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:

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

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293912
maximal normal subgroup of	C₂×#72293912
maximal quotient of	C₂×#72293941

GAP

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

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

C2×#3071131 #72294538