#193352

order	512
nilpotency class	3
isoclinism	isoclinic class of #18562
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#385
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

873 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 100548);