C₂×#7708710 #72256232

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

Aliases

direct product

C₂×#7708710

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

115345 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 7 generators on 22 points:

(7,8,10,12)(9,11,13,14)(16,19)(17,21);(15,16,18,19)(17,20,21,22);(4,6)(9,13)(11,14);(1,2);(9,13)(11,14)(17,21)(20,22);(3,4,5,6)(7,9)(8,11)(10,13)(12,14)(15,17,18,21)(16,20,19,22);(7,8)(9,14)(10,12)(11,13)(15,16)(17,22)(18,19)(20,21)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232033, C₂²×#7708710, #72329238, …
maximal normal subgroup of	C₂×#72232033, C₂²×#7708710, #72329238, …
maximal quotient of	C₂×#72233524, C₂×#72233525, C₂×#72233577, …

GAP

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

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

C2×#7708710 #72256232