#118404

order	512
nilpotency class	3
isoclinism	isoclinic class of #118395
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

606 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)(3,8)(4,7)(5,6)(9,10,14,18,15,19,13,20)(11,21,24,32,25,36,23,16)(12,26,28,34,30,17,29,35)(22,33,27,40,38,37,39,31);(1,2,5,8,4,7,3,6)(9,11,14,24,15,25,13,23)(10,16,20,36,19,32,18,21)(12,27,29,22,30,39,28,38)(17,33,35,40,26,37,34,31);(1,2,4,7)(3,8,5,6)(9,12,15,30)(10,17,19,26)(11,22,25,38)(13,28,14,29)(16,31,32,40)(18,34,20,35)(21,37,36,33)(23,27,24,39)

Polycyclic presentation:

2^9|a^2=g,b^2=dh,c^2=h,d^2=h,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,c^b=cf,f^b=fhi,d^c=di,f^c=fh,g^c=ghi

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, 25600);