C₂×#8757 #190163

order	512
nilpotency class	3
isoclinism	isoclinic class of #8746
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#8757

Subgroups

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

1194 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 1194 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) = 128

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

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 97359);