#97345

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, prime radical	C₂³	#379
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

590 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 100 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 — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=h,e^2=hi,f^2=g,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, 4541);