C₂×#4380242 #72294457

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄²

Aliases

direct product

C₂×#4380242

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

54633 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:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72251002
maximal normal subgroup of	C₂×#72251002
maximal quotient of	C₂×#72293957, C₂×#72293943

GAP

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

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

C2×#4380242 #72294457