#195476

order	512
nilpotency class	3
isoclinism	isoclinic class of #18746
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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×#386
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1279 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=dh,f^a=fi,e^b=eh,f^b=fh,g^b=gh,e^c=eh,f^c=fhi,g^c=gh

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 102672);