#120877

order	512
nilpotency class	3
isoclinism	isoclinic class of #120874
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	C₂²⋊Q₈
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:

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

Polycyclic presentation:

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

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, 28073);