C₂×#8059032 #72332250

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

Aliases

direct product

C₂×#8059032

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

At least 9972 conjugacy classes of subgroups; only the 31789 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 128

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

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

Conjugacy classes & character table

190 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323293
maximal normal subgroup of	C₂×#72323293
maximal quotient of	C₂×#72323384

GAP

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

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

C2×#8059032 #72332250