#107799

order	512
nilpotency class	3
isoclinism	isoclinic class of #9303
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂⁴	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1125 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=h,e^2=i,f^2=hi,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,d^b=di,e^c=ei,f^c=fi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 14995);