C₂×#10586743 #72299993

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁶

Aliases

direct product

C₂×#10586743

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂⁷
commutator, Frattini, cosocle	C₂²	C₂⁸
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂² ▷ C₁

At least 6752 conjugacy classes of subgroups; only the 417533 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 8 generators on 26 points:

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

Conjugacy classes & character table

280 conjugacy classes. The character table is too large to display.

Relationships

maximal quotient of

C₂×#72238233, C₂×#72297795

GAP

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

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

C2×#10586743 #72299993