#187900

order	512
nilpotency class	3
isoclinism	isoclinic class of #187866
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2226 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,e^2=i,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,e^b=ei,f^b=fhi,g^b=ghi,e^c=ehi,f^c=fh,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, 95096);