C₂×#7905877 #72304501

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

Aliases

direct product

C₂×#7905877

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 6422 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)(15,16,22,19)(17,23,29,25)(18,20,28,30)(21,24,26,27);(8,11)(9,13)(15,17)(16,23)(18,26)(19,25)(20,27)(21,28)(22,29)(24,30);(8,11)(12,14)(15,18)(16,20)(17,21)(19,30)(22,28)(23,24)(25,27)(26,29);(8,11)(9,13)(15,19,22,16)(17,25,29,23)(18,30,28,20)(21,27,26,24);(4,6)(8,11)(12,14)(15,18)(16,20)(17,26)(19,30)(21,29)(22,28)(23,27)(24,25);(3,4)(5,6)(7,8,10,11)(9,12,13,14)(15,20,22,30)(16,18,19,28)(17,27,29,24)(21,23,26,25);(7,9,10,13)(8,12,11,14)(15,21,22,26)(16,24,19,27)(17,28,29,18)(20,23,30,25)

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

C2×#7905877 #72304501