C₂×#2621398 #72337881

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

Aliases

direct product

C₂×#2621398

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

36197 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)(7,8)(9,13)(10,11)(12,14)(15,16)(17,20)(18,22)(19,21)(23,24)(25,29)(26,27)(28,30);(3,4,6,5)(8,11)(9,14)(15,17,19,22)(16,20,21,18)(24,27)(28,29);(4,5)(9,14)(12,13)(17,22)(18,20)(23,24,26,27)(25,28,30,29);(4,5)(7,8)(9,13)(10,11)(12,14)(15,16)(17,20)(18,22)(19,21)(23,24)(25,28)(26,27)(29,30);(1,2);(3,5,6,4)(7,9)(8,12)(10,14)(11,13)(15,18,19,20)(16,17,21,22)(23,25)(24,28)(26,30)(27,29)

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232289, C₂×#72232299, C₂×#72232297
maximal normal subgroup of	C₂×#72232289, C₂×#72232299, C₂×#72232297

GAP

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

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

C2×#2621398 #72337881