C₂×#1300082 #72337913

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

Aliases

direct product

C₂×#1300082

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

34451 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:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232357, C₂×#72232339, C₂×#72232297
maximal normal subgroup of	C₂×#72232357, C₂×#72232339, C₂×#72232297

GAP

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

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

C2×#1300082 #72337913