#115671

order	512
nilpotency class	3
isoclinism	isoclinic class of #115666
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

920 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)(4,7,5,8)(9,10)(11,19)(12,23,25,16)(13,26)(14,18)(15,21)(17,29)(20,37,39,36)(22,34)(24,33,28,35)(27,31)(30,40,32,38);(2,7)(3,8)(4,5)(9,11,14,22)(10,15,18,31)(12,20,25,39)(13,27,29,21)(16,32,23,30)(17,34,26,19)(24,40,28,38)(33,37,35,36);(1,3,6,2)(4,8,5,7)(9,12,14,25)(10,16,18,23)(11,20,22,39)(13,28,29,24)(15,30,31,32)(17,35,26,33)(19,36,34,37)(21,40,27,38)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,d^b=dh,f^b=fhi,d^c=dh,f^c=fhi,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, 22867);