#107676

order	512
nilpotency class	3
isoclinism	isoclinic class of #107658
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

764 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:

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

Polycyclic presentation:

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

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, 14872);