C₂×#7663385 #72298985

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂×#7663385

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁶
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

90041 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 30 points:

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72297863
maximal normal subgroup of	C₂×#72297863
maximal quotient of	C₂×#72298097

GAP

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

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

C2×#7663385 #72298985