#98464

order	512
nilpotency class	3
isoclinism	isoclinic class of #97316
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

655 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 655 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=h,e^2=hi,f^2=h,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=ei,f^b=fgh,d^c=di,e^c=ehi,f^c=fhi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5660);