C₂×#8915 #193020

order	512
nilpotency class	3
isoclinism	isoclinic class of #1655
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#8915

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1006 conjugacy classes of subgroups; only the 298 normal subgroups are shown.

normal subgroups only

This group has 1006 conjugacy classes of subgroups — too many to draw. Only its 298 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) = 38
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 38 points:

(1,2);(3,4,6,5)(7,8,13,18)(9,19,10,21)(11,17,12,16)(14,22,15,20)(24,28)(26,33)(29,37)(30,34)(31,35)(36,38);(3,5,6,4)(7,9,13,10)(8,14,18,15)(11,22,12,20)(16,21,17,19)(23,24,27,30)(25,29,32,38)(26,34,33,28)(31,36,35,37);(7,10,13,9)(8,15,18,14)(11,22,12,20)(16,21,17,19)(23,25,27,32)(24,29,30,38)(26,35,33,31)(28,36,34,37)

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, 100216);