#195391

order	512
nilpotency class	3
isoclinism	isoclinic class of #18284
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	#60940	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1376 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,f^a=fhi,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fh,g^c=gi

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, 102587);