C₂×#7691008 #72279705

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

Aliases

direct product

C₂×#7691008

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 6128 conjugacy classes of subgroups; only the 32149 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:

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

C2×#7691008 #72279705