#268718

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	#25745	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 939 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^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, 175914);