#92822

order	512
nilpotency class	4
isoclinism	isoclinic class of #1347
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#102
commutator	C₂×C₈	C₂×C₁₆
Frattini, cosocle	C₂×C₈²	C₂²
socle, prime radical	C₂³	D₄⋊C₈
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₈² ▷ C₂×C₄² ▷ C₂² ▷ C₁

203 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 203 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 2 generators on 40 points:

(1,2)(3,7,8,5)(4,6)(10,14)(11,17)(12,19)(15,24)(16,23)(18,21)(20,22)(25,26,27,30,28,31,33,36,29,32,34,37,35,38,39,40);(1,3)(2,5)(4,8)(6,7)(9,10,13,16)(11,18,20,24)(12,21,19,15)(14,22,23,17)

Polycyclic presentation:

2^9|a^2=d,b^2=h,c^2=fh,d^2=g,e^2=h,f^2=h,g^2=i,b^a=bc,c^a=ce,e^a=eh,f^a=fh,c^b=cf,d^b=defh,e^b=eh,f^b=fh

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 18);