C₂×#7692223 #72279871

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

Aliases

direct product

C₂×#7692223

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 6078 conjugacy classes of subgroups; only the 32189 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

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

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

Conjugacy classes & character table

226 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72262511
maximal normal subgroup of	C₂×#72262511

GAP

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

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

C2×#7692223 #72279871