#120715

order	512
nilpotency class	3
isoclinism	isoclinic class of #118505
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#388
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1442
prime radical	C₂⁴	C₄⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

467 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)(6,7)(9,10,14,18,15,20,13,19)(11,21,23,16,25,32,24,33)(12,26,28,36,29,35,30,17)(22,34,39,40,27,31,38,37);(1,2)(3,5)(4,6)(7,8)(9,11,13,24,15,25,14,23)(10,16,19,21,20,33,18,32)(12,27,28,38,29,22,30,39)(17,34,26,40,36,31,35,37);(2,6)(4,8)(5,7)(9,12,13,28,15,29,14,30)(10,17,18,35,20,36,19,26)(11,22,23,38,25,27,24,39)(16,31,21,37,33,34,32,40)

Polycyclic presentation:

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

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, 27911);