C₂×#7904709 #72304378

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

Aliases

direct product

C₂×#7904709

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 10138 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)(15,16,21,19)(17,22,20,25)(18,23,26,27)(24,29,28,30);(15,17)(16,22)(18,24)(19,25)(20,21)(23,29)(26,28)(27,30);(9,13)(12,14)(17,20)(22,25)(24,28)(29,30);(7,8,10,11)(9,12,13,14)(18,26)(23,27)(24,28)(29,30);(4,6)(9,13)(12,14)(15,18)(16,23)(17,24)(19,27)(20,28)(21,26)(22,29)(25,30);(7,8)(9,14)(10,11)(12,13)(15,19,21,16)(17,25,20,22)(18,23,26,27)(24,29,28,30);(3,4)(5,6)(7,9,10,13)(8,12,11,14)(15,20,21,17)(16,22,19,25)(18,24,26,28)(23,30,27,29)

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

C2×#7904709 #72304378