C₂×#123492 #72235166

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₄³ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

direct product

C₂×#123492

Subgroups

	subgroup	G/H
center	C₂³×C₄	#120
commutator	C₂×C₄	C₂×C₄³
Frattini, cosocle	C₂⁴×C₄	C₂⁴
socle	C₂⁴	#385
prime radical	C₂⁶	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

14229 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 26 points:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72235189, #72239949
maximal normal subgroup of	#72235189, #72239949
maximal quotient of	#72235282, #72239957

GAP

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

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

C2×#123492 #72235166