#131499

order	512
nilpotency class	4
isoclinism	isoclinic class of #131224
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

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

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

normal subgroups only

This group has 539 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=hi,c^2=g,d^2=h,e^2=i,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,e^b=egi,f^b=fgh,d^c=dg,f^c=fgi,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, 38695);