#120865

order	512
nilpotency class	3
isoclinism	isoclinic class of #10374
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#393
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1442
prime radical	C₂⁴	C₂×C₄⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

521 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,d^2=h,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,c^b=cf,d^b=di,f^c=fi,g^c=gi

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, 28061);