C₂×#7919685 #72304485

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

Aliases

direct product

C₂×#7919685

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 10148 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)(17,23,27,21)(18,19,25,24)(26,28,29,30);(15,17)(16,23)(18,26)(19,28)(20,21)(22,27)(24,30)(25,29);(3,4)(5,6)(7,8)(9,12)(10,11)(13,14)(15,18)(16,24)(17,26)(19,20)(21,28)(22,25)(23,30)(27,29);(3,4)(5,6)(7,8,10,11)(9,12,13,14)(16,20)(17,27)(18,25)(28,30);(4,6)(8,11)(12,14)(15,19)(16,25)(17,28)(18,20)(21,26)(22,24)(23,29)(27,30);(3,4)(5,6)(7,8)(9,14)(10,11)(12,13)(15,20)(16,22)(17,23)(18,19)(21,27)(24,25)(26,30)(28,29);(7,9,10,13)(8,12,11,14)(15,21,22,23)(16,17,20,27)(18,28,25,30)(19,29,24,26)

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

C2×#7919685 #72304485