C₂×#8657241 #72324538

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

Aliases

direct product

C₂×#8657241

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 10536 conjugacy classes of subgroups; only the 29589 normal subgroups are shown.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

178 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232069, C₂×#72232290
maximal normal subgroup of	C₂×#72232069, C₂×#72232290
maximal quotient of	C₂×#72232154, C₂×#72232157

GAP

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

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

C2×#8657241 #72324538