#317529

order	512
nilpotency class	3
isoclinism	isoclinic class of #317499
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

540 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 224725);