C₂×#7685078 #72279880

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²⁰

Aliases

direct product

C₂×#7685078

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

119947 conjugacy classes of subgroups.

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:

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

Conjugacy classes & character table

226 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72279669
maximal normal subgroup of	C₂×#72279669

GAP

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

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

C2×#7685078 #72279880