#111255

order	512
nilpotency class	3
isoclinism	isoclinic class of #111253
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1438
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

868 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=i,e^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,c^b=cf,d^b=di,e^b=ei,g^b=ghi,d^c=dh,e^c=eh,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, 18451);