#319406

order	512
nilpotency class	3
isoclinism	isoclinic class of #298795
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

735 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)(9,10,14,22,13,17,12,16)(11,18,20,15,19,24,21,23)(25,26,30,38,29,34,28,33)(27,31,35,32,37,40,36,39);(1,3,4,8)(2,5,6,7)(11,19)(15,23)(18,24)(20,21)(25,27,30,35,29,37,28,36)(26,31,38,32,34,40,33,39);(9,11)(10,15)(12,20)(13,19)(14,21)(16,24)(17,23)(18,22)(25,27,29,37)(26,32,34,39)(28,35,30,36)(31,33,40,38);(2,6)(5,7)(9,12,13,14)(10,16,17,22)(11,20,19,21)(15,24,23,18)(25,28,29,30)(26,33,34,38)(27,35,37,36)(31,32,40,39)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 226602);