#111809

order	512
nilpotency class	3
isoclinism	isoclinic class of #111807
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	#60559	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

917 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)(9,10,14,19,15,18,13,28)(11,20,24,36,22,16,23,31)(12,17,26,35,27,33,29,34)(21,30,25,40,38,37,39,32);(1,2)(3,7)(4,6)(5,8)(9,11)(10,16)(12,25)(13,24)(14,23)(15,22)(17,32)(18,20)(19,36)(21,26)(27,39)(28,31)(29,38)(30,34)(33,40)(35,37);(9,12)(10,17)(11,21)(13,29)(14,26)(15,27)(16,30)(18,33)(19,35)(20,37)(22,38)(23,39)(24,25)(28,34)(31,40)(32,36)

Polycyclic presentation:

2^9|a^2=g,d^2=i,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=di,e^b=eh,f^b=fh,g^b=ghi,f^c=fh

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, 19005);