C₂×#7670285 #72309035

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

Aliases

direct product

C₂×#7670285

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

75873 conjugacy classes of subgroups.

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:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72308003, C₂×#72308109
maximal normal subgroup of	C₂×#72308003, C₂×#72308109

GAP

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

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

C2×#7670285 #72309035