C₂×#7919711 #72304502

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

Aliases

direct product

C₂×#7919711

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 6382 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)(15,16,22,18)(17,23,26,27)(19,24,29,20)(21,28,30,25);(8,11)(9,13)(15,17)(16,23)(18,27)(19,21)(20,25)(22,26)(24,28)(29,30);(9,13)(12,14)(17,26)(21,30)(23,27)(25,28);(8,11)(9,13)(15,18,22,16)(17,27,26,23)(19,20,29,24)(21,25,30,28);(4,6)(8,11)(12,14)(15,19)(16,24)(17,21)(18,20)(22,29)(23,28)(25,27)(26,30);(3,4)(5,6)(7,8,10,11)(9,12,13,14)(15,20)(16,19)(17,28)(18,29)(21,27)(22,24)(23,30)(25,26);(7,9,10,13)(8,12,11,14)(15,21)(16,25)(17,29)(18,28)(19,26)(20,27)(22,30)(23,24)

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

C2×#7919711 #72304502