C₂×#188071 #72233293

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹²

Aliases

direct product

C₂×#188071

Subgroups

	subgroup	G/H
center	C₂²×C₄	#379
commutator	C₂⁴	C₂⁴×C₄
Frattini, cosocle	C₂³×C₄	C₂⁵
socle	C₂³	C₂×#379
prime radical	C₂³×#525	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

40397 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 26 points:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72214228, C₂×#72214665, C₂×#72213729
maximal normal subgroup of	C₂×#72214228, C₂×#72214665, C₂×#72213729
maximal quotient of	C₂×#72292877

GAP

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

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

C2×#188071 #72233293