C₂×#1126526 #72296393

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

Aliases

direct product

C₂×#1126526

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

34103 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 30 points:

(3,4)(5,6)(7,8)(9,13)(10,11)(12,14)(15,16)(17,21)(18,22)(19,20)(24,25)(28,29);(3,4,6,5)(9,14)(12,13)(16,20)(18,21)(23,24,27,25)(26,28,30,29);(4,5)(8,11)(9,14);(3,4,6,5)(7,8)(9,13)(10,11)(12,14)(15,17)(16,21)(18,20)(19,22)(23,25,27,24)(26,29,30,28);(3,5,6,4)(7,9)(8,12)(10,14)(11,13)(15,18,19,21)(16,22,20,17)(23,26,27,30)(24,28,25,29);(1,2)

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232907
maximal normal subgroup of	C₂×#72232907
maximal quotient of	C₂×#72278198

GAP

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

gap> G := Group((3,4)(5,6)(7,8)(9,13)(10,11)(12,14)(15,16)(17,21)(18,22)(19,20)(24,25)(28,29), (3,4,6,5)(9,14)(12,13)(16,20)(18,21)(23,24,27,25)(26,28,30,29), (4,5)(8,11)(9,14), (3,4,6,5)(7,8)(9,13)(10,11)(12,14)(15,17)(16,21)(18,20)(19,22)(23,25,27,24)(26,29,30,28), (3,5,6,4)(7,9)(8,12)(10,14)(11,13)(15,18,19,21)(16,22,20,17)(23,26,27,30)(24,28,25,29), (1,2));

C2×#1126526 #72296393