C₂×#7670317 #72309206

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

Aliases

direct product

C₂×#7670317

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 6250 conjugacy classes of subgroups; only the 32157 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:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72308109
maximal normal subgroup of	C₂×#72308109

GAP

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

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

C2×#7670317 #72309206