#121764

order	512
nilpotency class	3
isoclinism	isoclinic class of #112888
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1445
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

487 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=deh,d^2=i,e^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,c^b=cf,d^b=di,e^b=ei,g^b=ghi,d^c=dh,e^c=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28960);