C₂×#7942165 #72304397

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

Aliases

direct product

C₂×#7942165

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 6004 conjugacy classes of subgroups; only the 31893 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)(15,16,21,22)(17,23,25,29)(18,20,24,19)(26,28,30,27);(7,8,10,11)(9,12,13,14)(16,22)(17,25)(19,20)(26,30);(9,13)(12,14)(15,17)(16,23)(18,26)(19,27)(20,28)(21,25)(22,29)(24,30);(7,8,10,11)(9,12,13,14)(15,18,21,24)(16,19,22,20)(17,26,25,30)(23,27,29,28);(4,6)(7,8,10,11)(9,14,13,12)(15,19,21,20)(16,24,22,18)(17,27,25,28)(23,30,29,26);(7,8)(9,14)(10,11)(12,13)(15,20,21,19)(16,18,22,24)(17,28,25,27)(23,26,29,30);(3,4)(5,6)(7,9,10,13)(8,12,11,14)(15,17)(16,23)(18,26)(19,27)(20,28)(21,25)(22,29)(24,30)

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

C2×#7942165 #72304397