C₂×#1083751 #72294476

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

Aliases

direct product

C₂×#1083751

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

59691 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)(13,16)(14,15)(17,18)(19,20)(21,25)(22,24)(23,26);(13,17)(16,18)(19,21,22,26)(20,23,24,25);(4,6)(7,8,9,10)(12,15)(13,17)(20,24)(21,26);(3,4)(5,6)(7,8)(9,10)(11,12)(13,16)(14,15)(17,18)(19,20)(21,23)(22,24)(25,26);(11,13)(12,16)(14,17)(15,18)(19,21)(20,25)(22,26)(23,24);(1,2)

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72251002
maximal normal subgroup of	C₂×#72251002
maximal quotient of	C₂×#72293973, C₂×#72293986, C₂×#72293936, …

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

C2×#1083751 #72294476