#172478

order	512
nilpotency class	3
isoclinism	isoclinic class of #172137
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

804 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gi,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,d^b=dgh,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 79674);