C₂×#7942031 #72304465

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

Aliases

direct product

C₂×#7942031

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 6416 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)(15,16,21,19)(17,22,25,28)(18,23,27,29)(20,24,26,30);(4,6)(7,8,10,11)(9,12,13,14)(15,17)(16,22)(18,26)(19,28)(20,27)(21,25)(23,30)(24,29);(4,6)(8,11)(12,14)(15,18)(16,23)(17,20)(19,29)(21,27)(22,24)(25,26)(28,30);(4,6)(7,8,10,11)(9,12,13,14)(16,19)(17,25)(18,27)(24,30);(3,4,5,6)(8,11)(12,14)(15,18)(16,23)(17,26)(19,29)(20,25)(21,27)(22,30)(24,28);(7,8)(9,14)(10,11)(12,13)(15,19)(16,21)(17,22)(18,23)(20,30)(24,26)(25,28)(27,29);(7,9,10,13)(8,12,11,14)(15,20,21,26)(16,24,19,30)(17,27,25,18)(22,29,28,23)

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

C2×#7942031 #72304465