C₂×#8232071 #72300101

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸

Aliases

direct product

C₂×#8232071

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 6306 conjugacy classes of subgroups; only the 29541 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 26 points:

(3,4)(5,9)(6,10)(7,8)(11,12)(13,15)(14,16)(17,18)(20,25)(21,23);(11,13)(12,15)(14,17)(16,18)(19,20)(21,22)(23,26)(24,25);(3,4,7,8)(5,6,10,9)(13,17)(15,18)(19,21,24,23)(20,22,25,26);(5,10)(6,9)(11,12,14,16)(13,18,17,15)(19,22)(20,21)(23,25)(24,26);(3,5,7,10)(4,6,8,9)(11,12)(13,15)(14,16)(17,18)(19,23,24,21)(20,26,25,22);(3,6)(4,5)(7,9)(8,10)(11,13)(12,15)(14,17)(16,18)(19,20)(21,26)(22,23)(24,25);(1,2)

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72233585
maximal normal subgroup of	C₂×#72233585

GAP

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

gap> G := Group((3,4)(5,9)(6,10)(7,8)(11,12)(13,15)(14,16)(17,18)(20,25)(21,23), (11,13)(12,15)(14,17)(16,18)(19,20)(21,22)(23,26)(24,25), (3,4,7,8)(5,6,10,9)(13,17)(15,18)(19,21,24,23)(20,22,25,26), (5,10)(6,9)(11,12,14,16)(13,18,17,15)(19,22)(20,21)(23,25)(24,26), (3,5,7,10)(4,6,8,9)(11,12)(13,15)(14,16)(17,18)(19,23,24,21)(20,26,25,22), (3,6)(4,5)(7,9)(8,10)(11,13)(12,15)(14,17)(16,18)(19,20)(21,26)(22,23)(24,25), (1,2));

C2×#8232071 #72300101