C₂×#7918405 #72304408

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

Aliases

direct product

C₂×#7918405

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 6604 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)(8,11)(12,14)(15,16)(17,20)(18,26)(19,23)(21,28)(22,24)(25,30)(27,29);(7,8,10,11)(9,12,13,14)(15,17,19,24)(16,20,23,22)(18,27,28,25)(21,30,26,29);(9,13)(12,14)(17,24)(20,22)(25,27)(29,30);(7,8,10,11)(9,12,13,14)(18,28)(21,26)(25,27)(29,30);(4,6)(9,13)(12,14)(15,18)(16,21)(17,25)(19,28)(20,29)(22,30)(23,26)(24,27);(7,8)(9,12)(10,11)(13,14)(15,16)(17,20)(18,21)(19,23)(22,24)(25,29)(26,28)(27,30);(3,4)(5,6)(7,9)(8,12)(10,13)(11,14)(15,17)(16,22)(18,25)(19,24)(20,23)(21,30)(26,29)(27,28)

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

C2×#7918405 #72304408