C₂×#8742 #190071

order	512
nilpotency class	3
isoclinism	isoclinic class of #8738
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#8742

Subgroups

	subgroup	G/H
center, socle	C₂³	#381
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1238 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 1238 conjugacy classes of subgroups — too many to draw. Only its 258 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 97267);