#120855

order	512
nilpotency class	3
isoclinism	isoclinic class of #120853
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

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

453 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28051);