C₂×#2171279 #72296775

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁰×C₄²

Aliases

direct product

C₂×#2171279

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

45507 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 30 points:

(3,4,5,6)(7,8,10,12)(9,11,14,13)(15,16)(17,21)(18,20)(19,22)(25,29)(28,30);(15,17)(16,19)(18,22)(20,21)(23,24)(25,28)(26,27)(29,30);(3,4)(5,6)(16,20)(19,21)(24,27)(28,30);(3,4)(5,6)(7,9)(8,11)(10,14)(12,13)(15,17)(16,19)(18,22)(20,21)(23,25)(24,28)(26,29)(27,30);(3,4)(5,6)(7,8)(9,13)(10,12)(11,14)(15,16)(17,19)(18,20)(21,22)(23,24)(25,28)(26,27)(29,30);(1,2)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72278344
maximal normal subgroup of	C₂×#72278344
maximal quotient of	C₂×#72278515

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((3,4,5,6)(7,8,10,12)(9,11,14,13)(15,16)(17,21)(18,20)(19,22)(25,29)(28,30), (15,17)(16,19)(18,22)(20,21)(23,24)(25,28)(26,27)(29,30), (3,4)(5,6)(16,20)(19,21)(24,27)(28,30), (3,4)(5,6)(7,9)(8,11)(10,14)(12,13)(15,17)(16,19)(18,22)(20,21)(23,25)(24,28)(26,29)(27,30), (3,4)(5,6)(7,8)(9,13)(10,12)(11,14)(15,16)(17,19)(18,20)(21,22)(23,24)(25,28)(26,27)(29,30), (1,2));

C2×#2171279 #72296775