C₂×#3876711 #72294517

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

C₂×#3876711

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

32475 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:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293937, C₂×#72232681
maximal normal subgroup of	C₂×#72293937, C₂×#72232681
maximal quotient of	C₂×#72293954, C₂×#72293957

GAP

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

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

C2×#3876711 #72294517