#320027

order	512
nilpotency class	3
isoclinism	isoclinic class of #299067
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
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

780 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)(9,10,14,17,13,16,21,24)(11,12,19,20,18,15,23,22)(25,26,30,39,29,34,28,33)(27,35,36,32,37,31,38,40);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,16)(13,18)(14,19)(17,22)(20,24)(21,23)(25,27,28,38,29,37,30,36)(26,31,33,32,34,35,39,40);(9,12,13,15)(10,11,16,18)(14,22,21,20)(17,23,24,19)(25,27)(26,32)(28,36)(29,37)(30,38)(31,33)(34,40)(35,39);(2,6)(5,7)(10,16)(11,18)(17,24)(19,23)(25,28,29,30)(26,33,34,39)(27,36,37,38)(31,40,35,32)

Polycyclic presentation:

2^9|a^2=fi,b^2=egh,c^2=hi,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^b=dh,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 227223);