C₂×#7905733 #72304439

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

Aliases

direct product

C₂×#7905733

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 6592 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) = 128

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

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

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

C2×#7905733 #72304439