C₂×#7940421 #72304491

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

Aliases

direct product

C₂×#7940421

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 10124 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)(8,11)(9,13)(15,16,22,20)(17,23,25,21)(18,19,28,30)(24,26,27,29);(15,17)(16,23)(18,27)(19,29)(20,21)(22,25)(24,28)(26,30);(8,11)(12,14)(15,18)(16,19)(17,24)(20,30)(21,29)(22,28)(23,26)(25,27);(3,4)(5,6)(7,8,10,11)(9,12,13,14)(16,20)(17,25)(18,28)(26,29);(3,4,5,6)(7,8)(9,12)(10,11)(13,14)(15,19,22,30)(16,18,20,28)(17,26,25,29)(21,27,23,24);(3,4)(5,6)(7,8)(9,14)(10,11)(12,13)(15,20)(16,22)(17,23)(18,19)(21,25)(24,29)(26,27)(28,30);(7,9,10,13)(8,12,11,14)(15,21,22,23)(16,17,20,25)(18,29,28,26)(19,24,30,27)

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

C2×#7940421 #72304491