C₂×#10576928 #72231509

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂×#10576928

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂⁴×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁷
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#3444 ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

68337 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231575, C₂×#72213937, #72249556, …
maximal normal subgroup of	#72231575, C₂×#72213937, #72249556, …
maximal quotient of	C₂×#72237160, #72249556, D₃×#10576928, …
automorphism group of	#1382, C₂×#359, C₂×C₃×SD₃₂

GAP

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

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

C2×#10576928 #72231509