C₂×#7940277 #72304461

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

Aliases

direct product

C₂×#7940277

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 9826 conjugacy classes of subgroups; only the 31853 normal subgroups are shown.

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:

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

Conjugacy classes & character table

190 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72304329
maximal normal subgroup of	C₂×#72304329

GAP

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

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

C2×#7940277 #72304461