#256119

order	512
nilpotency class	3
isoclinism	isoclinic class of #246847
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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 ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

772 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,23)(20,22)(21,24)(25,26,29,33)(27,31,36,40)(28,32,30,37)(34,38,39,35);(2,5)(6,8)(9,11)(10,15)(12,20)(13,19)(14,21)(16,22)(17,23)(18,24)(25,27,29,36)(26,31,33,40)(28,35,30,38)(32,34,37,39);(1,3,4,7)(2,6,5,8)(10,16)(11,19)(12,13)(17,18)(20,21)(22,23)(25,26)(27,34)(28,32)(29,33)(30,37)(31,38)(35,40)(36,39);(2,5)(6,8)(9,12,14,13)(10,16,18,17)(11,20,21,19)(15,22,24,23)(25,28,29,30)(26,32,33,37)(27,35,36,38)(31,34,40,39)

Polycyclic presentation:

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

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, 163315);