C₂×#10586740 #72298802

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

Aliases

direct product

C₂×#10586740

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

477193 conjugacy classes of subgroups.

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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#10586740
maximal normal subgroup of	C₂²×#10586740
maximal quotient of	C₂×#72297477, C₂×#72297217, C₂×#72233749, …

GAP

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

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

C2×#10586740 #72298802