#268754

order	512
nilpotency class	3
isoclinism	isoclinic class of #268627
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3498	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

954 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 954 conjugacy classes of subgroups — too many to draw. Only its 144 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 — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fh,c^2=hi,d^2=i,e^2=i,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,f^c=fhi,g^c=gi,e^d=ei,f^d=fhi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 175950);