#120815

order	512
nilpotency class	3
isoclinism	isoclinic class of #120812
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1442
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

508 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=d,d^2=i,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=di,f^b=fi,g^b=ghi,e^c=eh,f^c=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28011);