C₂×#4045781 #72337966

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄

Aliases

direct product

C₂×#4045781

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

36573 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)(8,11)(12,14)(15,16,19,18)(17,21,22,20)(23,24,26,27)(25,28,29,30);(3,4,6,5)(8,11)(9,13)(17,22)(20,21)(24,27)(25,29);(3,4)(5,6)(8,11)(12,14)(16,18)(17,22)(23,24,26,27)(25,28,29,30);(4,5)(7,8)(9,14)(10,11)(12,13)(15,17)(16,20)(18,21)(19,22)(23,24)(25,28)(26,27)(29,30);(1,2);(3,5,6,4)(7,9)(8,12)(10,13)(11,14)(15,18,19,16)(17,21,22,20)(23,25)(24,28)(26,29)(27,30)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232283, C₂×#72232334, C₂×#72232298
maximal normal subgroup of	C₂×#72232283, C₂×#72232334, C₂×#72232298

GAP

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

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

C2×#4045781 #72337966