#131385

order	512
nilpotency class	4
isoclinism	isoclinic class of #131212
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1790
commutator	C₂×C₄⋊C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

670 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=h,e^2=i,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,d^b=dh,e^b=egi,f^b=fgh,d^c=dg,f^c=fg,e^d=ei,f^d=fi,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 38581);