C₂×#8141378 #72324608

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

Aliases

direct product

C₂×#8141378

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

68937 conjugacy classes of subgroups.

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232069, C₂×#72241227
maximal normal subgroup of	C₂×#72232069, C₂×#72241227
maximal quotient of	C₂×#72232098, C₂×#72232102

GAP

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

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

C2×#8141378 #72324608