C₂×#187811 #72233735

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰×C₄

Aliases

direct product

C₂×#187811

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#379
commutator	C₂²×C₄	C₂⁴×C₄
Frattini, cosocle	C₂×C₄²	C₂⁵
prime radical	C₂³×#535	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

39481 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);(4,7)(5,9)(13,21)(14,15)(17,25)(18,20)(22,26)(23,24);(11,12)(13,22)(14,18)(15,20)(16,19)(17,23)(21,26)(24,25);(3,4,5,8,6,7,9,10)(11,13)(12,17,19,22)(14,21)(15,24)(16,23)(18,25,20,26);(11,14,16,15)(12,18,19,20)(13,21,23,24)(17,25,22,26)

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72213377, C₂×#72213685, C₂×#72213738, …
maximal normal subgroup of	C₂×#72213377, C₂×#72213685, C₂×#72213738, …

GAP

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

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

C2×#187811 #72233735