C₂×#7631581 #72334354

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²¹

Aliases

direct product

C₂×#7631581

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁶
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

308413 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 7 generators on 26 points:

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

Conjugacy classes & character table

244 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72249537, #72337233
maximal normal subgroup of	C₂×#72249537, #72337233
maximal quotient of	C₂×#72256588, C₂×#72261376, C₂×#72240153, …

GAP

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

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

C2×#7631581 #72334354