C₂×#4606857 #72338051

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂⁹×C₄³

Aliases

direct product

C₂×#4606857

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

33705 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:

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

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232330, C₂×#72232365
maximal normal subgroup of	C₂×#72232330, C₂×#72232365

GAP

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

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

C2×#4606857 #72338051