C₂×#7918415 #72304346

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

Aliases

direct product

C₂×#7918415

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 6092 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)(3,4)(5,6)(7,8)(13,14)(15,16)(17,25)(18,28)(19,29)(20,24)(21,23)(22,26)(27,30);(7,9,8,11)(10,13,12,14)(15,17)(16,22)(18,27)(19,30)(20,28)(21,26)(23,25)(24,29);(7,9,8,11)(10,14,12,13)(16,23)(17,26)(19,28)(24,27);(7,9,8,11)(10,13,12,14)(15,18,21,29)(16,19,23,28)(17,27,26,24)(20,22,30,25);(5,6)(10,12)(13,14)(15,19)(16,18)(17,20)(21,28)(22,24)(23,29)(25,27)(26,30);(7,9)(8,11)(10,14)(12,13)(15,19)(16,18)(17,20)(21,28)(22,24)(23,29)(25,27)(26,30);(3,5)(4,6)(7,10,8,12)(9,13,11,14)(15,20)(16,24)(17,19)(18,22)(21,30)(23,27)(25,29)(26,28)

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

C2×#7918415 #72304346