#116375

order	512
nilpotency class	3
isoclinism	isoclinic class of #116371
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	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

645 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,f^a=fhi,c^b=cf,e^b=ei,d^c=dh,e^c=ei,f^c=fh,g^c=gi

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, 23571);