#274649

order	512
nilpotency class	3
isoclinism	isoclinic class of #268891
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2856	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

686 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=ei,c^2=hi,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,g^b=gi,d^c=di,f^c=fh,g^c=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 181845);