C₂×#2542881 #72337990

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

Aliases

direct product

C₂×#2542881

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

38121 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)(7,8,11,9)(10,13,14,12)(16,17)(18,22)(23,24)(25,29)(26,27)(28,30);(3,4,6,5)(8,9)(10,14)(16,17)(20,21)(24,27)(28,29);(4,5)(8,9)(12,13)(15,16,19,17)(18,20,22,21)(23,24,26,27)(25,28,30,29);(4,5)(7,9,11,8)(10,12,14,13)(15,17,19,16)(18,20,22,21)(23,24)(25,29)(26,27)(28,30);(1,2);(3,5,6,4)(7,10,11,14)(8,12,9,13)(15,18,19,22)(16,20,17,21)(23,25,26,30)(24,28,27,29)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232289, C₂×#72232295, C₂×#72232308
maximal normal subgroup of	C₂×#72232289, C₂×#72232295, C₂×#72232308

GAP

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

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

C2×#2542881 #72337990